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{"name": "Simon's copy of Gauss elimination to solve a system of linear equations.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "unitTests": [], "customName": "", "prompt": "\n        <p>Re-arrange the rows so that the third row becomes the first row, the first the second and the second the third.<br/><strong>WHY?</strong> Choose one of the following:<br/>[[0]]</p>\n      <p>Now write down the entries of the matrix you will use for Gaussian Elimination, remember to include the constants as the last column.</p>\n      <table>\n      <tbody>\n      <tr>\n      <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      <td>[[1]]</td>\n      <td>[[2]]</td>\n      <td>[[3]]</td>\n      <td>[[4]]</td>\n      <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      </tr>\n      <tr>\n      <td>[[5]]</td>\n      <td>[[6]]</td>\n      <td>[[7]]</td>\n      <td>[[8]]</td>\n      </tr>\n      <tr>\n      <td>[[9]]</td>\n      <td>[[10]]</td>\n      <td>[[11]]</td>\n      <td>[[12]]</td>\n      </tr>\n      </tbody>\n      </table>\n        ", "showFeedbackIcon": true, "variableReplacements": [], "gaps": [{"scripts": {}, "showCorrectAnswer": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "choices": ["<p>To make sure that there is a 1 in the first row, first column position.</p>", "<p>Because you always do this.</p>", "<p>Why not.</p>", "<p>I don't know.</p>"], "type": "1_n_2", "shuffleChoices": true, "displayType": "radiogroup", "unitTests": [], "minMarks": 0, "customName": "", "maxMarks": 1, "distractors": ["", "", "", ""], "showFeedbackIcon": true, "variableReplacements": [], "matrix": [1, 0, 0, 0], 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"variableReplacementStrategy": "originalfirst", "type": "gapfill", "unitTests": [], "customName": "", "prompt": "\n      \n      \n      <p>Now introduce zeros in the first column below the first entry by adding:<br/>[[0]] times the first row to the second row and<br/>[[1]] times the first row to the third row to get the matrix:</p>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\"> \\[\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> $\\var{b}$ </td>\n      \n      <td> $\\var{b*a-b}$ </td>\n      \n      <td> $\\var{c3}$ </td>\n      \n      <td rowspan=\"3\"> \\[\\left) \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$ </td>\n      \n      <td> [[2]] </td>\n      \n      <td> [[3]] </td>\n      \n      <td> [[4]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> [[5]] </td>\n      \n      <td> [[6]] </td>\n      \n      <td> [[7]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <p>Next multiply the second row by [[8]] to get a 1 in the second entry in the second row.</p>\n      \n      \n        ", "showFeedbackIcon": true, "variableReplacements": [], "gaps": [{"scripts": {}, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "marks": 0.8, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "type": "numberentry", "unitTests": [], "maxValue": "{-a}", "customName": "", "correctAnswerFraction": false, "showFeedbackIcon": true, "variableReplacements": [], "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "useCustomName": false, "minValue": "{-a}", "showFractionHint": true, 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"extendBaseMarkingAlgorithm": true}, {"scripts": {}, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "marks": 0.8, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "type": "numberentry", "unitTests": [], "maxValue": "{-1}", "customName": "", "correctAnswerFraction": false, "showFeedbackIcon": true, "variableReplacements": [], "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "useCustomName": false, "minValue": "{-1}", "showFractionHint": true, "extendBaseMarkingAlgorithm": true}], "customMarkingAlgorithm": "", "useCustomName": false, "sortAnswers": false, "extendBaseMarkingAlgorithm": true}, {"scripts": {}, "showCorrectAnswer": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "unitTests": [], "customName": "", "prompt": "\n      \n      \n      <p>Note that you should have multiplied the second row by a suitable number to get a $1$ in the second entry in the second row.<br/>In this part we introduce a $0$ in the second column below the second entry in the second column by adding:<br/>[[0]] times the second row to the third row to get the matrix:</p>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\"> \\[\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> $\\var{b}$ </td>\n      \n      <td> $\\var{b*a-b}$ </td>\n      \n      <td> $\\var{c3}$ </td>\n      \n      <td rowspan=\"3\"> \\[\\left) \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> [[1]] </td>\n      \n      <td> [[2]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> [[3]] </td>\n      \n      <td> [[4]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <p>From this you should find:</p>\n      \n      \n      \n      <p>$z=\\;\\;$[[5]]</p>\n      \n      \n        ", "showFeedbackIcon": true, "variableReplacements": [], "gaps": [{"scripts": {}, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "marks": 0.8, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "type": "numberentry", "unitTests": [], "maxValue": "{-b*c+a*b*c}", "customName": "", "correctAnswerFraction": false, "showFeedbackIcon": true, "variableReplacements": [], "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "useCustomName": false, "minValue": "{-b*c+a*b*c}", "showFractionHint": true, "extendBaseMarkingAlgorithm": true}, {"scripts": {}, "correctAnswerStyle": "plain", "showCorrectAnswer": 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"en", "si-en"], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "useCustomName": false, "minValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "showFractionHint": true, "extendBaseMarkingAlgorithm": true}, {"scripts": {}, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "marks": 2, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "type": "numberentry", "unitTests": [], "maxValue": "{z}", "customName": "", "correctAnswerFraction": false, "showFeedbackIcon": true, "variableReplacements": [], "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "useCustomName": false, "minValue": "{z}", "showFractionHint": true, "extendBaseMarkingAlgorithm": true}], "customMarkingAlgorithm": "", "useCustomName": false, "sortAnswers": false, "extendBaseMarkingAlgorithm": true}, {"scripts": {}, "showCorrectAnswer": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "unitTests": [], "customName": "", "prompt": "\n      \n      \n      <p>From the second row of the reduced matrix you find an equation involving only $y$ and $z$ and using your value for $z$ we find:</p>\n      \n      \n      \n      <p>$y=\\;\\;$[[0]]</p>\n      \n      \n      \n      <p>Then using the first row we have the equation :<br/>\\[\\simplify[all]{x+ {b}y+{b*a-b}z={c3}}\\]</p>\n      \n      \n      \n      <p>Using this you can now find $x$:</p>\n      \n      \n      \n      <p>$x=\\;\\;$[[1]]</p>\n      \n      \n        ", "showFeedbackIcon": true, "variableReplacements": [], "gaps": [{"scripts": {}, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "marks": 3, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "type": "numberentry", "unitTests": [], "maxValue": "{y}", "customName": "", "correctAnswerFraction": false, "showFeedbackIcon": true, "variableReplacements": [], "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], 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"anything", "description": "", "name": "c3", "definition": "random(1..3)"}, "z": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "z", "definition": "c1+c2*c*(b-a*b)+c3*c*(a^2*b-a-a*b)"}, "y": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "y", "definition": "a*c3-c2-a*z"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a", "definition": "random(2..6)"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c1", "definition": "random(1..5)"}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c2", "definition": "random(1..3)"}, "x": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "x", "definition": "c3-(b*a-b)*z-b*y"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b", "definition": "random(2..6)"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c", "definition": "random(1..3)"}}, "name": "Simon's copy of Gauss elimination to solve a system of linear equations.", "variable_groups": [], "ungrouped_variables": ["a", "c", "b", "c3", "c2", "y", "x", "c1", "z"], "statement": "<p>Solve the system of equations using Gauss Elimination <br/>\\[\\begin{eqnarray*} &amp;\\var{a}x&amp;+\\;&amp;\\var{a*b-1}y&amp;+\\;\\var{a^2*b-a-a*b}z&amp;=&amp;\\var{c2}\\\\ &amp;\\var{a*c}x&amp;+\\;&amp;\\var{c*b}y&amp;+\\;z&amp;=&amp;\\var{c1}\\\\ &amp;x&amp;+\\;&amp;\\var{b}y&amp;+\\;\\var{b*a-b}z&amp;=&amp;\\var{c3} \\end{eqnarray*} \\] <br/><strong>Part a)</strong> Rearrange the order of the equations and represent this as a system of equations using a matrix.<br/><strong>Part b)</strong> Introduce zeros in the first column using the first row.<br/><strong>Part c)</strong> Introduce zeros in the second coumn below the second entry in the second row using the second row. <br/>Also need to solve for $z$ using the last row of the reduced matrix.<br/><strong>Part d)</strong> Solve for $y$ and $x$ using the second and first rows of the reduced matrix.</p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "advice": "<p>We can write the system of linear equations in matrix form as follows:</p>\n<p>\\[\\begin{pmatrix} \\var{a} &amp; \\var{a*b-1}&amp;\\var{a^2*b-a-a*b} &amp;\\vdots&amp; \\var{c2}\\\\ \\var{a*c}&amp;\\var{c*b}&amp;1&amp;\\vdots&amp;\\var{c1}\\\\ 1&amp;\\var{b}&amp;\\var{b*a-b}&amp;\\vdots&amp;\\var{c3} \\end{pmatrix} \\]</p>\n<p>Rearranging the rows to give us a 1 as the upper left value gives:</p>\n<p>\\[\\begin{pmatrix} 1&amp;\\var{b}&amp;\\var{b*a-b}&amp;\\vdots&amp;\\var{c3}\\\\\\var{a} &amp; \\var{a*b-1}&amp;\\var{a^2*b-a-a*b} &amp;\\vdots&amp; \\var{c2}\\\\ \\var{a*c}&amp;\\var{c*b}&amp;1&amp;\\vdots&amp;\\var{c1} \\end{pmatrix} \\]</p>\n<p>Subtracting $\\var{a}$ lots of row 1 from row 2 gives:&nbsp;</p>\n<p>\\[\\begin{pmatrix} 1&amp;\\var{b}&amp;\\var{b*a-b}&amp;\\vdots&amp;\\var{c3}\\\\ 0 &amp; \\var{-1}&amp;\\var{-a} &amp;\\vdots&amp; \\var{c2-c3*a}\\\\ \\var{a*c}&amp;\\var{c*b}&amp;1&amp;\\vdots&amp;\\var{c1} \\end{pmatrix} \\]</p>\n<p>Subtracting $\\var{a*c}$ lots of row 1 from row 3 gives:&nbsp;</p>\n<p>\\[\\begin{pmatrix} 1&amp;\\var{b}&amp;\\var{b*a-b}&amp;\\vdots&amp;\\var{c3}\\\\ 0 &amp; \\var{-1}&amp;\\var{-a} &amp;\\vdots&amp; \\var{c2-c3*a}\\\\ 0 &amp;\\var{c*b-a*c*b}&amp;\\var{1-a^2*c*b+a*c*b}&amp;\\vdots&amp;\\var{c1-a*c*c3} \\end{pmatrix} \\]</p>\n<p>Now our first column is in the desired form.</p>\n<p>We can obtain a leading 1 in the second row by dividing row 2 by -1:</p>\n<p>\\[\\begin{pmatrix} 1&amp;\\var{b}&amp;\\var{b*a-b}&amp;\\vdots&amp;\\var{c3}\\\\ 0 &amp; \\var{1}&amp;\\var{a} &amp;\\vdots&amp; \\var{-c2+c3*a}\\\\0&amp;\\var{c*b-a*c*b}&amp;\\var{1-a^2*c*b+a*c*b}&amp;\\vdots&amp;\\var{c1-a*c*c3} \\end{pmatrix} \\]</p>\n<p>Adding $\\var{-(c*b-a*c*b)}$ lots of row 2 to row 3 gives:</p>\n<p>\\[\\begin{pmatrix} 1&amp;\\var{b}&amp;\\var{b*a-b}&amp;\\vdots&amp;\\var{c3}\\\\ 0 &amp; \\var{1}&amp;\\var{a} &amp;\\vdots&amp; \\var{-c2+c3*a}\\\\0&amp;0&amp;1&amp;\\vdots&amp;\\var{c1-a*c*c3-(c*b-a*c*b)*(-c2+c3*a)} \\end{pmatrix} \\]</p>\n<p>The matrix is now in upper diagonal form so we can quickly solve the system of equations as follows:</p>\n<p>The third row gives $z =&nbsp;\\var{c1-a*c*c3-(c*b-a*c*b)*(-c2+c3*a)}$</p>\n<p>The second row gives $y+\\var{a}z=\\var{-c2+c3*a}$, and hence $y=&nbsp;\\var{-c2+c3*a}-\\var{a}\\times&nbsp;\\var{c1-a*c*c3-(c*b-a*c*b)*(-c2+c3*a)} =&nbsp;\\var{-c2+c3*a-a*(c1-a*c*c3-(c*b-a*c*b)*(-c2+c3*a))}$</p>\n<p>The first row gives $x+\\var{b}y+\\var{b*a-b}z = \\var{c3}$ and hence $x= \\var{c3}-\\var{b} \\times&nbsp;\\var{-c2+c3*a-a*(c1-a*c*c3-(c*b-a*c*b)*(-c2+c3*a))} - \\var{b*a-b} \\times \\var{c1-a*c*c3-(c*b-a*c*b)*(-c2+c3*a)}=\\var{x}$</p>", "extensions": [], "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}