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{"name": "Gaussian Elimination", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"gaps": [{"matrix": [1, 0, 0, 0], "displayType": "radiogroup", "marks": 0, "showCorrectAnswer": true, "choices": ["<p>To make the arithmetic easier for ourselves even though it is not the best pivot.</p>", "<p>To make the Gaussian Elimination algorithm numerically stable you should always do this.</p>", "<p>The best pivot in this question is 1.</p>", "<p>I don't know.</p>"], "scripts": {}, "type": "1_n_2", "shuffleChoices": true, "displayColumns": 0, "maxMarks": 1, "distractors": ["", "", "", ""], "minMarks": 0}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "1", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "1", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{b}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{b}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{b*a-b}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{b*a-b}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{c3}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{c3}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{a}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{a}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{a*b-1}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{a*b-1}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{a^2*b-a-a*b}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{a^2*b-a-a*b}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{c2}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{c2}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{a*c}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{a*c}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{b*c}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{b*c}", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "1", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "1", "marks": 0.1}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{c1}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{c1}", "marks": 0.1}], "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "prompt": "<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 rowspan=\"3\">\\[\\left| \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</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>"}, {"gaps": [{"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{-a}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{-a}", "marks": 0.4}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{-a*c}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{-a*c}", "marks": 0.4}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{-1}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{-1}", "marks": 0.3}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{-a}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{-a}", "marks": 0.3}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{c2-a*c3}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{c2-a*c3}", "marks": 0.3}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{c*b-c*b*a}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{c*b-c*b*a}", "marks": 0.3}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{-a^2*b*c+1+a*b*c}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{-a^2*b*c+1+a*b*c}", "marks": 0.3}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{c1-a*c*c3}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{c1-a*c*c3}", "marks": 0.3}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{-1}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{-1}", "marks": 0.4}], "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "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      <td rowspan=\"3\">\\[\\left| \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\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        \n      \n        "}, {"gaps": [{"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{-b*c+a*b*c}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{-b*c+a*b*c}", "marks": 0.4}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{a}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{a}", "marks": 0.4}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{a*c3-c2}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{a*c3-c2}", "marks": 0.4}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{1}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{1}", "marks": 0.4}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "marks": 0.4}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{z}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{z}", "marks": 1}], "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "prompt": "<p>(Note that from the last part, you may have multiplied the second row by a suitable number to get a $1$ in the second entry in the second row.)</p>\n        <p><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        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>$\\var{1}$</td>\n        <td>$\\var{b}$</td>\n        <td>$\\var{b*a-b}$</td>\n        <td rowspan=\"3\">\\[\\left| \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>$\\var{c3}$</td>\n        <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$\\var{0}$</td>\n        <td>$\\var{1}$</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        </tr>\n        <tr>\n        <td>$\\var{0}$</td>\n        <td>$\\var{0}$</td>\n        <td>[[3]]</td>\n        <td>[[4]]</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>From this you should find:</p>\n        <p>$z=\\;\\;$[[5]]</p>"}, {"gaps": [{"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{y}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{y}", "marks": 1.5}, {"scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "{x}", "showCorrectAnswer": true, "type": "numberentry", "allowFractions": false, "minValue": "{x}", "marks": 1.3}], "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "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        \n      \n        "}], "name": "Gaussian Elimination", "metadata": {"notes": "<p><strong>30/03/2015:</strong> NB for simplest algebra $a_{11}$ is made equal to 1, not stable pivoting. Changed the MCQ to highlight this.</p>\n<p><strong>17/02/2013:</strong> JDP added augmented matrix bars</p>\n<p><strong>5/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>Changed some of the grammar in the instructions.</p>\n<p><strong>9/07/2012:</strong></p>\n<p>Corrected tags. Note that tags are not case sensitive.</p>\n<p>Added more tags.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.</p>"}, "statement": "<p>Solve the system of equations using Gaussian 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>", "functions": {}, "type": "question", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"c1": {"templateType": "anything", "name": "c1", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}, "b": {"templateType": "anything", "name": "b", "description": "", "group": "Ungrouped variables", "definition": "random(2..6)"}, "c3": {"templateType": "anything", "name": "c3", "description": "", "group": "Ungrouped variables", "definition": "random(1..3)"}, "c2": {"templateType": "anything", "name": "c2", "description": "", "group": "Ungrouped variables", "definition": "random(1..3)"}, "c": {"templateType": "anything", "name": "c", "description": "", "group": "Ungrouped variables", "definition": "random(1..3)"}, "x": {"templateType": "anything", "name": "x", "description": "", "group": "Ungrouped variables", "definition": "c3-(b*a-b)*z-b*y"}, "y": {"templateType": "anything", "name": "y", "description": "", "group": "Ungrouped variables", "definition": "a*c3-c2-a*z"}, "a": {"templateType": "anything", "name": "a", "description": "", "group": "Ungrouped variables", "definition": "random(2..6)"}, "z": {"templateType": "anything", "name": "z", "description": "", "group": "Ungrouped variables", "definition": "c1+c2*c*(b-a*b)+c3*c*(a^2*b-a-a*b)"}}, "preamble": {"js": "", "css": ""}, "tags": ["Gauss", "Gaussian elimination", "Linear equations", "MA4100", "Solving equations", "columns", "echelon form", "gauss", "gaussian elimination", "linear algebra", "linear equations", "ma4100", "matrix", "reduced matrix", "row operators", "rows", "solving a system of linear equations", "solving equations"], "variable_groups": [], "ungrouped_variables": ["a", "b", "c", "c1", "c2", "c3", "y", "x", "z"], "advice": "<p>Look at the revealed answers for this question. All the information needed is there.</p>", "question_groups": [{"pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered", "questions": []}], "showQuestionGroupNames": false, "contributors": [{"name": "James Denholm-Price", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/17/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "James Denholm-Price", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/17/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}