{"id":13227,"date":"2025-05-01T07:01:43","date_gmt":"2025-05-01T07:01:43","guid":{"rendered":"https:\/\/www.hnjournal.net\/?page_id=13227"},"modified":"2025-05-01T07:01:45","modified_gmt":"2025-05-01T07:01:45","slug":"6-5-14","status":"publish","type":"page","link":"https:\/\/www.hnjournal.net\/en\/6-5-14\/","title":{"rendered":""},"content":{"rendered":"<div class=\"journal-article\" style=\"margin-bottom: 20px;\"><h3 style='text-align: left; font-family:Times New Roman;'>Solving Linear Systems with non-integer coefficients by using some soft ware<\/h3><h4 style='text-align: right; font-family:Simplified Arabic;'><\/h4><p style='text-align: left; font-weight:bold;'>Dr. Abbas Abdelaziz Gumma Mahmoud<sup>1<\/sup><\/p><div style='direction: ltr; text-align: left; font-size:12px; line-height:1.5;'><p><sup>1<\/sup> Department of Mathematics, Al-Zaeem Al-Azhari University, Sudan.<\/p><\/div><p style='text-align:left;'><strong>DOI:<\/strong> <a href='https:\/\/doi.org\/https:\/\/doi.org\/10.53796\/hnsj65\/14' target='_blank' rel='noopener'>https:\/\/doi.org\/10.53796\/hnsj65\/14<\/a><\/p><p style='text-align: left;'><strong>Arabic Scientific Research Identifier:<\/strong> <a href='https:\/\/arsri.org\/10000\/65\/14' target='_blank' rel='noopener'>https:\/\/arsri.org\/10000\/65\/14<\/a><\/p><p style='text-align: left;'><strong>Volume (6) Issue (5). Pages:<\/strong> 157 - 165<\/p><p style='text-align: left;'><strong>Received at:<\/strong> 2025-04-07 | <strong>Accepted at:<\/strong> 2025-04-15 | <strong>Published at:<\/strong> 2025-05-01<\/p><p><a href='\/volume6\/issue5\/6-5-14.pdf' target='_blank' rel='noopener' style='background-color:green;color:white;padding:10px 15px;text-decoration:none;border-radius:5px;'>Download PDF<\/a><\/p>\r\n<style>\r\n.hnsj-cite-btn{\r\n  display:inline-flex; gap:8px; align-items:center;\r\n  padding:10px 14px; border-radius:10px;\r\n  border:1px solid #0b5ed7; background:#0b5ed7; color:#fff;\r\n  cursor:pointer; font-weight:700;\r\n}\r\n.hnsj-cite-btn:hover{background:#084bb0;border-color:#084bb0}\r\n.hnsj-cite-note{display:block;margin-top:6px;font-size:13px;opacity:.85}\r\n\r\n.hnsj-modal-backdrop{\r\n  position:fixed; inset:0; background:rgba(0,0,0,.55);\r\n  display:none; z-index:99998;\r\n}\r\n.hnsj-modal{\r\n  position:fixed; left:50%; top:50%; transform:translate(-50%,-50%);\r\n  width:min(760px,94vw); background:#fff; border-radius:14px;\r\n  box-shadow:0 12px 35px rgba(0,0,0,.28);\r\n  display:none; z-index:99999; overflow:hidden;\r\n  border:1px solid rgba(0,0,0,.08);\r\n}\r\n\r\n.hnsj-modal-header{\r\n  display:flex; justify-content:space-between; align-items:center;\r\n  padding:14px 16px; border-bottom:1px solid #eee; background:#f8fafc;\r\n}\r\n.hnsj-modal-title{font-size:16px;font-weight:800;color:#111827}\r\n.hnsj-modal-close{\r\n  border:1px solid #d1d5db; background:#fff;\r\n  width:34px; height:34px; border-radius:10px;\r\n  font-size:18px; cursor:pointer; line-height:0; color:#111827;\r\n}\r\n.hnsj-modal-close:hover{background:#f3f4f6}\r\n\r\n.hnsj-tabs{\r\n  display:flex; gap:10px; padding:10px 16px;\r\n  border-bottom:1px solid #f0f0f0; justify-content:flex-end;\r\n}\r\n.hnsj-tab{\r\n  padding:10px 14px; border-radius:10px;\r\n  border:1px solid #cfcfcf; background:#f3f4f6;\r\n  cursor:pointer; font-weight:800; color:#111827;\r\n}\r\n.hnsj-tab:hover{background:#e5e7eb;border-color:#9ca3af}\r\n.hnsj-tab.active{\r\n  background:#0b5ed7; border-color:#0b5ed7; color:#fff;\r\n  box-shadow:0 2px 10px rgba(11,94,215,.18);\r\n}\r\n\r\n.hnsj-modal-body{padding:14px 16px}\r\n.hnsj-row{\r\n  display:flex; gap:10px; flex-wrap:wrap; align-items:center;\r\n  margin-bottom:10px; justify-content:flex-end;\r\n}\r\n.hnsj-select{\r\n  padding:10px 12px; border-radius:10px;\r\n  border:1px solid #cfcfcf; min-width:220px;\r\n  background:#fff; color:#111827; font-weight:700;\r\n}\r\n.hnsj-copy{\r\n  padding:10px 14px; border-radius:10px;\r\n  border:1px solid #0b5ed7; background:#0b5ed7; color:#fff;\r\n  cursor:pointer; font-weight:800;\r\n}\r\n.hnsj-copy:hover{background:#084bb0;border-color:#084bb0}\r\n\r\n.hnsj-textarea{\r\n  width:100%; min-height:130px; padding:12px;\r\n  border-radius:12px; border:1px solid #cfcfcf;\r\n  line-height:1.7; resize:vertical; color:#111827; background:#fff;\r\n}\r\n.hnsj-actions{display:flex; justify-content:space-between; align-items:center; margin-top:10px; gap:10px; flex-wrap:wrap;}\r\n.hnsj-dl{\r\n  padding:10px 14px;\r\n  border-radius:10px;\r\n  border:1px solid #0b5ed7;\r\n  background:#0b5ed7;\r\n  color:#fff;\r\n  cursor:pointer;\r\n  font-weight:800;\r\n}\r\n.hnsj-dl:hover{background:#084bb0;border-color:#084bb0}\r\n\/* Force the citation modal UI to be independent from site RTL\/LTR *\/\r\n.hnsj-modal,\r\n.hnsj-modal *{\r\n  direction: ltr;\r\n  text-align: left;\r\n}\r\n\r\n\/* Keep the header title readable *\/\r\n.hnsj-modal-header{\r\n  direction: ltr;\r\n}\r\n<\/style>\r\n\r\n<script>\r\n(function(){\r\n  function slugifyFileName(s){\r\n    return (s || 'citation')\r\n      .toString()\r\n      .trim()\r\n      .replace(\/^https?:\\\/\\\/\/i,'')\r\n      .replace(\/[^a-z0-9]+\/gi,'-')\r\n      .replace(\/-+\/g,'-')\r\n      .replace(\/^-|-$\/g,'')\r\n      .toLowerCase();\r\n  }\r\n\r\n  function downloadTextFile(filename, content, mime){\r\n    var blob = new Blob([content], { type: mime || 'text\/plain;charset=utf-8' });\r\n    var url = URL.createObjectURL(blob);\r\n    var a = document.createElement('a');\r\n    a.href = url;\r\n    a.download = filename;\r\n    document.body.appendChild(a);\r\n    a.click();\r\n    a.remove();\r\n    setTimeout(function(){ URL.revokeObjectURL(url); }, 500);\r\n  }\r\n\r\n  function splitAuthors(str){\r\n    if(!str) return [];\r\n    return str\r\n      .split(\/,|\u061b|\u060c|;|\\n\/g)\r\n      .map(s => s.trim())\r\n      .filter(Boolean);\r\n  }\r\n\r\n  function buildRIS(m, langKey){\r\n    const title   = (langKey === 'ar') ? (m.title_ar || m.title_en || '') : (m.title_en || m.title_ar || '');\r\n    const journal = (langKey === 'ar') ? (m.journal_ar || m.journal_en || '') : (m.journal_en || m.journal_ar || '');\r\n    const authors = splitAuthors((langKey === 'ar') ? (m.authors_ar || m.authors_en || '') : (m.authors_en || m.authors_ar || ''));\r\n\r\n    let ris = '';\r\n    ris += 'TY  - JOUR\\n';\r\n    if (title)   ris += 'TI  - ' + title + '\\n';\r\n    authors.forEach(a => { ris += 'AU  - ' + a + '\\n'; });\r\n    if (journal) ris += 'JO  - ' + journal + '\\n';\r\n    if (m.year)  ris += 'PY  - ' + m.year + '\\n';\r\n    if (m.volume)ris += 'VL  - ' + m.volume + '\\n';\r\n    if (m.issue) ris += 'IS  - ' + m.issue + '\\n';\r\n    if (m.doi)   ris += 'DO  - ' + m.doi + '\\n';\r\n    if (m.url)   ris += 'UR  - ' + m.url + '\\n';\r\n    ris += 'ER  - \\n';\r\n    return ris;\r\n  }\r\n\r\n  function buildBibTeX(m, langKey){\r\n    const title   = (langKey === 'ar') ? (m.title_ar || m.title_en || '') : (m.title_en || m.title_ar || '');\r\n    const journal = (langKey === 'ar') ? (m.journal_ar || m.journal_en || '') : (m.journal_en || m.journal_ar || '');\r\n    const authorStr = (langKey === 'ar') ? (m.authors_ar || m.authors_en || '') : (m.authors_en || m.authors_ar || '');\r\n\r\n    const keyBase = slugifyFileName(m.doi || title || 'hnsj');\r\n    const key = (keyBase || 'hnsj').slice(0, 40);\r\n\r\n    let bib = '';\r\n    bib += '@article{' + key + ',\\n';\r\n    if (title)      bib += '  title = {' + title + '},\\n';\r\n    if (authorStr)  bib += '  author = {' + authorStr + '},\\n';\r\n    if (journal)    bib += '  journal = {' + journal + '},\\n';\r\n    if (m.year)     bib += '  year = {' + m.year + '},\\n';\r\n    if (m.volume)   bib += '  volume = {' + m.volume + '},\\n';\r\n    if (m.issue)    bib += '  number = {' + m.issue + '},\\n';\r\n    if (m.doi)      bib += '  doi = {' + m.doi + '},\\n';\r\n    if (m.url)      bib += '  url = {' + m.url + '},\\n';\r\n    bib = bib.replace(\/,\\n$\/, '\\n');\r\n    bib += '}\\n';\r\n    return bib;\r\n  }\r\n\r\n  function initHnsjCite(root){\r\n    if(!root) return;\r\n\r\n    \/\/ Guard: do not bind twice\r\n    if (root.dataset.hnsjInited === '1') return;\r\n    root.dataset.hnsjInited = '1';\r\n\r\n    const openBtn = root.querySelector('[data-hnsj-open]');\r\n    const modal   = root.querySelector('[data-hnsj-modal]');\r\n    const back    = root.querySelector('[data-hnsj-backdrop]');\r\n    const closeBtn= root.querySelector('[data-hnsj-close]');\r\n\r\n    const tabs    = modal.querySelectorAll('.hnsj-tab');\r\n    const select  = modal.querySelector('[data-hnsj-style]');\r\n    const box     = modal.querySelector('[data-hnsj-box]');\r\n    const copyBtn = modal.querySelector('[data-hnsj-copy]');\r\n    const dlBtns  = modal.querySelectorAll('[data-hnsj-dl]');\r\n\r\n    let lang = 'en';\r\n    let citEn = {};\r\n    let citAr = {};\r\n    let meta = {};\r\n\r\n    function parseData(){\r\n      try{ citEn = JSON.parse(modal.getAttribute('data-cit-en') || '{}'); }catch(e){ citEn = {}; }\r\n      try{ citAr = JSON.parse(modal.getAttribute('data-cit-ar') || '{}'); }catch(e){ citAr = {}; }\r\n      try{ meta  = JSON.parse(modal.getAttribute('data-meta')  || '{}'); }catch(e){ meta = {}; }\r\n    }\r\n\r\n    function setLang(newLang){\r\n      lang = newLang;\r\n      tabs.forEach(t => t.classList.toggle('active', t.dataset.lang === lang));\r\n\r\n      const data = (lang === 'ar') ? citAr : citEn;\r\n      const styles = Object.keys(data);\r\n\r\n      select.innerHTML = '';\r\n      styles.forEach(st => {\r\n        const opt = document.createElement('option');\r\n        opt.value = st;\r\n        opt.textContent = st;\r\n        select.appendChild(opt);\r\n      });\r\n\r\n      box.dir = (lang === 'ar') ? 'rtl' : 'ltr';\r\n      box.style.textAlign = (lang === 'ar') ? 'right' : 'left';\r\n\r\n      if(styles.length){\r\n        select.value = styles[0];\r\n        box.value = data[styles[0]] || '';\r\n      } else {\r\n        box.value = '';\r\n      }\r\n    }\r\n\r\n    function openModal(){\r\n      parseData();\r\n      back.style.display = 'block';\r\n      modal.style.display = 'block';\r\n      document.body.style.overflow = 'hidden';\r\n      setLang('en');\r\n    }\r\n\r\n    function closeModal(){\r\n      back.style.display = 'none';\r\n      modal.style.display = 'none';\r\n      document.body.style.overflow = '';\r\n    }\r\n\r\n    function onStyleChange(){\r\n      const data = (lang === 'ar') ? citAr : citEn;\r\n      box.value = data[select.value] || '';\r\n    }\r\n\r\n    async function copyCitation(){\r\n      try{\r\n        await navigator.clipboard.writeText(box.value || '');\r\n      }catch(e){\r\n        box.focus(); box.select();\r\n        document.execCommand('copy');\r\n      }\r\n      const old = copyBtn.textContent;\r\n      copyBtn.textContent = (lang === 'ar') ? '\u062a\u0645 \u0627\u0644\u0646\u0633\u062e \u2713' : 'Copied \u2713';\r\n      setTimeout(()=> copyBtn.textContent = old, 1200);\r\n    }\r\n\r\n    openBtn.addEventListener('click', openModal);\r\n    closeBtn.addEventListener('click', closeModal);\r\n    back.addEventListener('click', closeModal);\r\n\r\n    tabs.forEach(t => t.addEventListener('click', ()=> setLang(t.dataset.lang)));\r\n    select.addEventListener('change', onStyleChange);\r\n    copyBtn.addEventListener('click', copyCitation);\r\n\r\n    dlBtns.forEach(btn => {\r\n      btn.addEventListener('click', function(){\r\n        parseData();\r\n        const kind = btn.getAttribute('data-hnsj-dl'); \/\/ ris or bib\r\n        const langKey = lang || 'en';\r\n        const base = slugifyFileName(meta.doi || meta.title_en || meta.title_ar || 'citation');\r\n\r\n        if (kind === 'ris') {\r\n          downloadTextFile(base + '.ris', buildRIS(meta, langKey), 'application\/x-research-info-systems;charset=utf-8');\r\n        } else {\r\n          downloadTextFile(base + '.bib', buildBibTeX(meta, langKey), 'application\/x-bibtex;charset=utf-8');\r\n        }\r\n      });\r\n    });\r\n\r\n    document.addEventListener('keydown', (e)=>{\r\n      if(e.key === 'Escape' && modal.style.display === 'block') closeModal();\r\n    });\r\n  }\r\n\r\n  function bootHnsjCite(){\r\n    document.querySelectorAll('[data-hnsj-cite-root]').forEach(initHnsjCite);\r\n  }\r\n\r\n  \/\/ Run now if DOM already ready, otherwise wait\r\n  if (document.readyState === 'loading') {\r\n    document.addEventListener('DOMContentLoaded', bootHnsjCite);\r\n  } else {\r\n    bootHnsjCite();\r\n  }\r\n\r\n  \/\/ Also handle late injected content (builders\/cache)\r\n  const obs = new MutationObserver(() => bootHnsjCite());\r\n  obs.observe(document.documentElement, { childList: true, subtree: true });\r\n\r\n})();\r\n<\/script>\r\n\r\n<div data-hnsj-cite-root id='hnsjCite_95222'>\r\n  <button type='button' class='hnsj-cite-btn' data-hnsj-open>\r\n    <span>\ud83d\udccc<\/span><span>Cite this article<\/span>\r\n  <\/button>\r\n\r\n  <div class='hnsj-modal-backdrop' data-hnsj-backdrop><\/div>\r\n\r\n  <div class='hnsj-modal' data-hnsj-modal data-meta='{&quot;authors_en&quot;:&quot;Dr. Abbas Abdelaziz Jumaa Mahmoud&quot;,&quot;authors_ar&quot;:&quot;Dr. Abbas Abdelaziz Gumma Mahmoud&quot;,&quot;title_en&quot;:&quot;Solving Linear Systems with non-integer coefficients by using some soft ware&quot;,&quot;title_ar&quot;:&quot;Solving Linear Systems with non-integer coefficients by using some soft ware&quot;,&quot;journal_en&quot;:&quot;Humanities &amp; Natural Sciences Journal&quot;,&quot;journal_ar&quot;:&quot;\u0645\u062c\u0644\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u0627\u0644\u0627\u0646\u0633\u0627\u0646\u064a\u0629 \u0648\u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629&quot;,&quot;year&quot;:&quot;2025&quot;,&quot;volume&quot;:&quot;6&quot;,&quot;issue&quot;:&quot;5&quot;,&quot;doi&quot;:&quot;https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;url&quot;:&quot;https:\\\/\\\/www.hnjournal.net\\\/ar\\\/6-5-14\\\/&quot;,&quot;published_at&quot;:&quot;2025-05-01&quot;}' data-cit-en='{&quot;APA&quot;:&quot;Mahmoud A. A. G. (2025). Solving Linear Systems with non-integer coefficients by using some soft ware. Humanities &amp; Natural Sciences Journal, 6(5). https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;Chicago&quot;:&quot;Mahmoud Abbas Abdelaziz Gumma. 2025. \\&quot;Solving Linear Systems with non-integer coefficients by using some soft ware.\\&quot; Humanities &amp; Natural Sciences Journal 6, no. 5. https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;Harvard&quot;:&quot;Mahmoud A. A. G. 2025. Solving Linear Systems with non-integer coefficients by using some soft ware. Humanities &amp; Natural Sciences Journal. [Internet] 2025-05-01. [Cited 2026-05-15]. 6(5). Available at: https:\\\/\\\/www.hnjournal.net\\\/ar\\\/6-5-14\\\/. https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;Vancouver&quot;:&quot;Mahmoud A. A. G. Solving Linear Systems with non-integer coefficients by using some soft ware. Humanities &amp; Natural Sciences Journal. [Internet]. 2025-05-01; 6(5). Available from: https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;IEEE&quot;:&quot;Mahmoud A. A. G, \\&quot;Solving Linear Systems with non-integer coefficients by using some soft ware,\\&quot; Humanities &amp; Natural Sciences Journal, vol. 6, no. 5, 2025. https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;MLA&quot;:&quot;Mahmoud Abbas Abdelaziz Gumma. \\&quot;Solving Linear Systems with non-integer coefficients by using some soft ware.\\&quot; Humanities &amp; Natural Sciences Journal, vol. 6, no. 5, 2025-05-01, https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;}' data-cit-ar='{&quot;APA&quot;:&quot;Mahmoud A. A. G. (2025). Solving Linear Systems with non-integer coefficients by using some soft ware. \u0645\u062c\u0644\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u0627\u0644\u0627\u0646\u0633\u0627\u0646\u064a\u0629 \u0648\u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629\u060c 6(5). https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;Chicago&quot;:&quot;Mahmoud Abbas Abdelaziz Gumma. 2025. \u00abSolving Linear Systems with non-integer coefficients by using some soft ware\u00bb. \u0645\u062c\u0644\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u0627\u0644\u0627\u0646\u0633\u0627\u0646\u064a\u0629 \u0648\u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629\u060c 6(5). https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;Harvard&quot;:&quot;Mahmoud A. A. G. Solving Linear Systems with non-integer coefficients by using some soft ware. \u0645\u062c\u0644\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u0627\u0644\u0627\u0646\u0633\u0627\u0646\u064a\u0629 \u0648\u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629. [\u0627\u0646\u062a\u0631\u0646\u062a] 2025-05-01. [\u062a\u0627\u0631\u064a\u062e \u0627\u0644\u0648\u0635\u0648\u0644 2026-05-15]. 6(5). \u0645\u062a\u0627\u062d \u0639\u0644\u0649: https:\\\/\\\/www.hnjournal.net\\\/ar\\\/6-5-14\\\/. https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;Vancouver&quot;:&quot;Mahmoud A. A. G. Solving Linear Systems with non-integer coefficients by using some soft ware. \u0645\u062c\u0644\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u0627\u0644\u0627\u0646\u0633\u0627\u0646\u064a\u0629 \u0648\u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629. [\u0627\u0646\u062a\u0631\u0646\u062a]. 2025-05-01\u061b 6(5). \u0645\u062a\u0627\u062d \u0645\u0646: https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;IEEE&quot;:&quot;Mahmoud A. A. G. \u00abSolving Linear Systems with non-integer coefficients by using some soft ware\u00bb. \u0645\u062c\u0644\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u0627\u0644\u0627\u0646\u0633\u0627\u0646\u064a\u0629 \u0648\u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629\u060c \u0645 6\u060c \u0639 5\u060c 2025. https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;,&quot;MLA&quot;:&quot;Mahmoud Abbas Abdelaziz Gumma. \u00abSolving Linear Systems with non-integer coefficients by using some soft ware\u00bb. \u0645\u062c\u0644\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u0627\u0644\u0627\u0646\u0633\u0627\u0646\u064a\u0629 \u0648\u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629\u060c \u0645 6\u060c \u0639 5\u060c 2025-05-01\u060c https:\\\/\\\/doi.org\\\/10.53796\\\/hnsj65\\\/14&quot;}'>\r\n    <div class='hnsj-modal-header'>\r\n    <div class='hnsj-modal-title'>Cite \/ \u0627\u0644\u0627\u0633\u062a\u0634\u0647\u0627\u062f<\/div>\r\n    <button class='hnsj-modal-close' type='button' data-hnsj-close aria-label='Close'>\u00d7<\/button>\r\n    <\/div>\r\n\r\n    <div class='hnsj-tabs'>\r\n      <button type='button' class='hnsj-tab active' data-lang='en'>English (Roman)<\/button>\r\n      <button type='button' class='hnsj-tab' data-lang='ar'>\u0627\u0644\u0639\u0631\u0628\u064a\u0629<\/button>\r\n    <\/div>\r\n\r\n    <div class='hnsj-modal-body'>\r\n      <div class='hnsj-row'>\r\n        <button type='button' class='hnsj-copy' data-hnsj-copy>Copy<\/button>\r\n        <select class='hnsj-select' data-hnsj-style><\/select>\r\n        <\/div>\r\n\r\n      <textarea class='hnsj-textarea' data-hnsj-box readonly><\/textarea>\r\n\r\n      <div class='hnsj-actions'>\r\n        <div style='display:flex; gap:10px; flex-wrap:wrap;'>\r\n          <button type='button' class='hnsj-dl' data-hnsj-dl='ris'>Download RIS<\/button>\r\n          <button type='button' class='hnsj-dl' data-hnsj-dl='bib'>Download BibTeX<\/button>\r\n        <\/div>\r\n      <\/div>\r\n    <\/div>\r\n  <\/div>\r\n<\/div>\r\n<p style='text-align:justify; direction:ltr;'><strong>Abstract:<\/strong> The present study aims to look for optimization programs to solve systems of linear equations and so these methods requiring less mathematical skills and effort mentally contributes to less than reliable in various applications for non-specialists in mathematics. And then compare this software to find the difference between them and the errors if it is existed. The case study involved a written system contained a non-integer coefficient to note the differences in accuracy of solutions. In this case identical solutions of all systems up to 10-6.<\/p><p style='text-align:left; direction:ltr;'><strong>Keywords: <\/strong> linear systems, optimization software, information technology, and life problems.<\/p><\/div>\n\n\n<p dir=\"ltr\" style=\"text-align: justify;\">.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><a id=\"post-13227-OLE_LINK20\"><\/a><strong>1. Introduction<\/strong>: In this paper we will try to take advantage of the great development in the field of software to employ them in the direction of building mathematical models and resolution.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">The aims look for optimization programs to solve systems of linear equations and so these methods requiring less mathematical skills and effort mentally contributes to less than reliable in various applications for non-specialists in mathematics. And then compare this software to find the difference between them and the errors if it is exist. The study helps to shorten the time in the solution of linear systems using some ready-made software with less effort and small errors. Also through the study note that can non-professionals in the field of mathematics to deal with linear systems.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>2. Study goal: <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">The present study aims to look for optimization programs to solve systems of linear equations and so these methods requiring less mathematical skills and effort mentally contributes to less than reliable in various applications for non-specialists in mathematics. And then compare this software to find the difference between them and the errors if it is exist.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>3. The problem of the study:<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Study the problem lies in the difficulty of solving systems of linear equations by standard methods so to find the exact solutions to these systems using ready-made software.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>4. The importance of the study:<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">The study helps to shorten the time in the solution of linear systems using some ready-made software with less effort and small errors. Also through the study note that can non-professionals in the field of mathematics to deal with linear systems.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>5. Case study :<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">In this case the coefficients of the variables are non-integer<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>9.9 x<sub>1<\/sub> \u2013 1.5 x<sub>2<\/sub> + 2.6 x<sub>3<\/sub> = 0<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>0.4 x<sub>1<\/sub> + 13.6 x<sub>2<\/sub> -4.2 x<sub>3<\/sub> = 8.2<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>0.7 x<sub>1<\/sub> + 0.4 x<sub>2<\/sub> + 7.1 x<sub>3<\/sub> = -1.3<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Solution of Case Study 2 manually by Iterative Method: <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Reduce the system to the normal form:<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>9.9 x<sub>1<\/sub> = 1.5 x<sub>2<\/sub> &#8211; 2.6 x<sub>3<\/sub> <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>13.6 x<sub>2<\/sub> = 8.2 &#8211; 0.4 x<sub>1 <\/sub>+ 4.2 x<sub>3<\/sub><\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>7.1 x<sub>3<\/sub> = -1.3 &#8211; 0.7 x<sub>1<\/sub> &#8211; 0.4 x<sub>2<\/sub><\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Or<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>x1 = x2 \u2013 x3<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>x2 = &#8211; x1 + x3<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>x3 = &#8211; x1 \u2013 x2 <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">\uf103 = ,<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Then write the system in the form<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">x = + \uf103 x<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">zero approximation:<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">x =<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">or:<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">=<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">First approximation:<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">= + x =<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Second approximation:<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">= + x =<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Third approximation:<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">= + x =<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Forth approximation:<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">= + x =<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">The following Table(1) shows the answers of case study 2 by<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Approximation method (Iterative Method):<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (1)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Approximation Method Answers<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Variable<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Value<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong> 0.1397<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong> 0.5288<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong> -0.2267<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Table (2) shows the verifying of the solutions.<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (2)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">verification of Approximation Method<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Equation<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Constant<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Substitution Value<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>The error<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.0004099999999<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.00040999999999<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.1997<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.000299999999997<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.30026<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.000259999999999<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>4.Solution of Case by using Excel Solver: <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">As shown in Figure (1), enter the system<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" width=\"411\" height=\"134\" class=\"wp-image-13292\" src=\"http:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-1.png\" srcset=\"https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-1.png 411w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-1-300x98.png 300w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-1-18x6.png 18w\" sizes=\"auto, (max-width: 411px) 100vw, 411px\" \/><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" width=\"665\" height=\"479\" class=\"wp-image-13293\" src=\"http:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-2.png\" srcset=\"https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-2.png 665w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-2-300x216.png 300w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-2-18x12.png 18w\" sizes=\"auto, (max-width: 665px) 100vw, 665px\" \/> Then the solution shows like in Figure (2).<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Figure (2): Excel Answer <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">The following Table (3) shows the answers of case by<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Excel Solver.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (3)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Answers of Excel<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Variable<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Value<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.139653690654187<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.528835594026146<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.226660820233984<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Table (4) shows the verifying of the solutions.<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (4)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">verification of Excel<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Equation<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Constant<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Substitution Value<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>The error<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>1.38 <\/strong>x <strong>10<sup>-08<\/sup><\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.0000000138289<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.200001<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.0000001<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>5. Solution of Case by using Lindo: <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" width=\"645\" height=\"324\" class=\"wp-image-13294\" src=\"http:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-3.png\" srcset=\"https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-3.png 645w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-3-300x151.png 300w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-3-18x9.png 18w\" sizes=\"auto, (max-width: 645px) 100vw, 645px\" \/> Figure (3) shows the entry of the Case :<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Figure (3): Lindo input <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" width=\"440\" height=\"176\" class=\"wp-image-13295\" src=\"http:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-4.png\" srcset=\"https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-4.png 440w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-4-300x120.png 300w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-4-18x7.png 18w\" sizes=\"auto, (max-width: 440px) 100vw, 440px\" \/> Then the solution appear as shows in Figure (4).<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Figure(4): Lindo Answer <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">The following Table (5) shows the answers of case by Lindo<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (5)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Answers of Lindo<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Variable<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Value<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.139654<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.528836<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.226661<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (6) shows the verifying of the solutions of Case using Lindo.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (6)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Case verification of Lindo solution<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Equation<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Constant<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Substitution Value<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>The error<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.000002<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.000002<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.2000074<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.0000074<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3000009<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.00000089999999989<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong> 6. Solution of Case Study using Maxima:<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong><img loading=\"lazy\" decoding=\"async\" width=\"319\" height=\"221\" class=\"wp-image-13296\" src=\"http:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-5.png\" srcset=\"https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-5.png 319w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-5-300x208.png 300w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-5-18x12.png 18w\" sizes=\"auto, (max-width: 319px) 100vw, 319px\" \/> <\/strong>Enter the system as seen in Figure (5)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Figure (5): Maxima input <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Figure (6): Maxima Answers <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" width=\"829\" height=\"70\" class=\"wp-image-13297\" src=\"http:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-6.png\" srcset=\"https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-6.png 829w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-6-300x25.png 300w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-6-768x65.png 768w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-6-18x2.png 18w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-6-820x70.png 820w\" sizes=\"auto, (max-width: 829px) 100vw, 829px\" \/> Then the solution appears as Figure (6)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">The following Table (7) shows the answers of case study 2 using<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Maxima.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (7)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Answers of Maxima<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Variable<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Value<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.13965367693931500000<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.52883552267193300000<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.22666081449666100000<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (8) shows the verifying of the solutions of Case using maxima.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (8)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">verification of Maxima solution<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Equation<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Constant<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Substitution Value<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>The error<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.19999999999999<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.000000000000001<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong> 6. Solution of Case Study by using SimSolve : <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" width=\"474\" height=\"271\" class=\"wp-image-13298\" src=\"http:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-7.png\" srcset=\"https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-7.png 474w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-7-300x172.png 300w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-7-18x10.png 18w\" sizes=\"auto, (max-width: 474px) 100vw, 474px\" \/> Figure (7) shows the input of the system in SimSolve<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Figure (7): SimSolve input <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" width=\"444\" height=\"366\" class=\"wp-image-13299\" src=\"http:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-8.png\" srcset=\"https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-8.png 444w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-8-300x247.png 300w, https:\/\/www.hnjournal.net\/wp-content\/uploads\/2025\/05\/word-image-13227-8-15x12.png 15w\" sizes=\"auto, (max-width: 444px) 100vw, 444px\" \/> Then the solution shows as Figure (8)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>Figure (8): SimSolve Answers <\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">The following Table (9) shows the answers by using SimSolve.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (9)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Answers of SimSolve<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Variable<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Value<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.139653676939315<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.528835522671934<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>X3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.226660814496661<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (10) shows the verifying of the solutions of Case 2 using SimSolve.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">Table (10)<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">verification of SimSolve solution<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\n<p><strong>Equation<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Constant<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Substitution Value<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>The error<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>8.2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation 3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-1.3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\">\u00a0<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">7. Conclusion : The case study involved a written system contained <br \/>a non-integer coefficient to note the differences in accuracy of solutions. <br \/>In this case identical solutions of all systems up to 10-6<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>In the case sudy represents a sample of:<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">non-integer coefficients, and thus may produce solutions not identical values. This case recorded the following :<\/p>\n<table dir=\"ltr\">\n<tbody>\n<tr>\n<td>\u00a0<\/td>\n<td colspan=\"6\">\n<p>Error<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\n<p><strong>Iterative<\/strong><\/p>\n<p><strong>Method<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Excel Solver<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>LINDO 6.1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>MATLAB<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Maxima<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>SimSolve<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation1<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.000409<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.00000001<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.000002<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.000409<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation2<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.0002999<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.0000001<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>-0.000007<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.0002999<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Equation3<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.0002599<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.00000089<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.0002599<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p><strong>Avg<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.000322<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.000003<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0.000322<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>0<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p dir=\"ltr\" style=\"text-align: justify;\">Avg = (\u2211 Error )\/number of Errors<\/p>\n<ul dir=\"ltr\" style=\"text-align: justify;\">\n<li>All the software in the match results untill 10<sup>-5<\/sup> , exept MATLAB match untill 10<sup>-3<\/sup> .<\/li>\n<li>MATLAB sofware and Sidel Method (manully) gives same solution .<\/li>\n<li>Since SimSolve uses one of the methods of elimation (Gauss-Jordan) so there are no errors in the solution to the lack of non-zero values in the coefficients matrix.<\/li>\n<\/ul>\n<p dir=\"ltr\" style=\"text-align: justify;\">By offering a solution in the Maxima it appeared to be running one of the iterative methods, which repeats and substitute until error is belong to zero. So nonexisting error.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\"><strong>References<\/strong><\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">1.A. Barry (2003). Computational methods in O.R using Lindo and Lingo (2003). Available at: <a href=\"http:\/\/www.abarry.ws\/books\/CalculationMethodsBook.pdf\">http:\/\/www.abarry.ws\/books\/CalculationMethodsBook.pdf<\/a>\/<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">2.Bosch, W. W., and Strickland, J. (1998). Systems of linear equations on a spreadsheet.\u00a0<em>Mathematics &amp; Computer Education, 32<\/em>, (1), 11-16.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">3.Camporesi, Roberto (2016). A Fresh Look at Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization. <em>International Journal of Mathematical Education in Science and Technology<\/em>, v47 n1 p82-99 2016.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">4.Carley, Holly (2014). On Solving Systems of Equations by Successive Reduction Using 2\u00d72 Matrices. <em>Australian Senior Mathematics Journal<\/em>, v28 n1 p43-56 2014.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">5.Deane Arganbright (n. d). Innovative Uses of Excel in Linear Algebra. Department of Mathematics and Computing Science Divine Word University, Papua New Guinea. Available online at: http:\/\/atcm. mathandtech. org\/EP2011 \/regular _papers \/ 3272011_19192.pdf<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">6.Edwards, Thomas G.; Chelst, Kenneth R.; Principato, Angela M. and Wilhelm, Thad L. (2015). Investigating Integer Restrictions in Linear Programming. <em>Mathematics Teacher<\/em>, v109 n2 p136-142 Sep 2015.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">7.El-Gebeily, M. and Yushau, B. (2008). Linear System of Equations, Matrix Inversion, and Linear Programming Using MS Excel.\u00a0<em>International Journal of Mathematical Education in Science and Technology, 39<\/em>(1), 83-94.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">8.Greg Harvey (2007). <em>Microsoft Office Excel 2007: All-in-one desk reference<\/em> .Wiley Publishing, Inc.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">9.Jerome, Lawrence (2012). Generalized assignment matrix methodology in linear programming. <em>International Journal for Technology in Mathematics Education<\/em>, v19 n1 p25-41 2012.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">10.LINDO User&#8217;s Manual (2003). LINDO SYSTEMS INC. Available online at: <a href=\"http:\/\/www.lindo.com\">http:\/\/www.lindo.com<\/a>.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">11.Macsyma system (2001). Maxima User Manual.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">12.Martin Golubitsky and Michal Dellnitz (1999). <em>Linear algebra &amp; differential equations using MATLAB.<\/em> An International Thomson Publisher Company.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">13.Mike Brookers (2011). <em>The Matrix Reference Manual<\/em>. Imperial College \u2013 London.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">14.N. I. Danilina, N. S. Dubrovskaya, O. P. Kvasha and G. L. Samirov (1988). <em>Computational Mathematics<\/em>. Mir Publisher.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">15.O. V. Manturov and N. M. Matveev (1989). <em>A Course of Higher Mathematics<\/em>. Mir Publisher.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">16.R. L. Burden and J. D. Faires (2011). <em>Numerical Analysis (9<sup>th<\/sup> ed.)<\/em>.Youngstown State University.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">17.Roanes-Lozano, Eugenio (2011). Some applications of algebraic system solving. <em>International Journal for Technology in Mathematics Education; 2011, Vol. 18<\/em> Issue 3, p149.<\/p>\n<p dir=\"ltr\" style=\"text-align: justify;\">18.Stephen Moffat (2011). Excel 2010 Part II. Stephen Moffat &amp; bookboon.com Available online at: http \/\/www.bookboon.com.<\/p>\n<h2 dir=\"ltr\" style=\"text-align: justify;\">\u00a0<\/h2>\n<p dir=\"ltr\" style=\"text-align: justify;\">\u00a0<\/p>","protected":false},"excerpt":{"rendered":"<p>. 1. Introduction: In this paper we will try to take advantage of the great development in the field of software to employ them in the direction of building mathematical [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_joinchat":[],"footnotes":""},"class_list":["post-13227","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/pages\/13227","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/comments?post=13227"}],"version-history":[{"count":2,"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/pages\/13227\/revisions"}],"predecessor-version":[{"id":13335,"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/pages\/13227\/revisions\/13335"}],"wp:attachment":[{"href":"https:\/\/www.hnjournal.net\/en\/wp-json\/wp\/v2\/media?parent=13227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}