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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": [{"advice": "<p>Look at the revealed answers for this question. All the information needed is there.</p>", "variable_groups": [], "ungrouped_variables": ["a", "b", "c", "c1", "c2", "c3", "y", "x", "z"], "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "statement": "<p>Solve the system of equations using Gaussian Elimination <br/>\\[\\begin{eqnarray} &amp;x&amp;+\\;&amp;\\var{b}y&amp;+\\;\\var{b*a-b}z&amp;=&amp;\\var{c3}\\\\ &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} \\end{eqnarray} \\] <br/><strong>Part a)</strong> Introduce zeros in the first column using the first row.<br/><strong>Part b)</strong>&nbsp;Introduce zeros in the second coumn below the second entry in the second row using the second row, then solve for $z$ using the last row of the reduced matrix.<br/><strong>Part c)</strong>&nbsp;Solve for $y$ and $x$ using the second and first rows of the reduced matrix.<strong>&nbsp;</strong></p>", "preamble": {"js": "", "css": ""}, "tags": [], "parts": [{"showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "scripts": {}, "prompt": "<p>Write the augmented matrix corresponding to this system of equations</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td rowspan=\"3\">\\[\\left| \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n<td>[[3]]</td>\n<td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n</tr>\n<tr>\n<td>[[4]]</td>\n<td>[[5]]</td>\n<td>[[6]]</td>\n<td>[[7]]</td>\n</tr>\n<tr>\n<td>[[8]]</td>\n<td>[[9]]</td>\n<td>[[10]]</td>\n<td>[[11]]</td>\n</tr>\n</tbody>\n</table>", "gaps": 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"gapfill"}, {"showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "scripts": {}, "prompt": "<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<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>[[2]]</td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n</tr>\n<tr>\n<td>$\\var{0}$</td>\n<td>[[5]]</td>\n<td>[[6]]</td>\n<td>[[7]]</td>\n</tr>\n</tbody>\n</table>\n<p>Next multiply the second row by [[8]] to get a 1 in the second entry in the second row.</p>", "gaps": [{"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{-a}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{-a}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{-a*c}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", 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"notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{-a^2*b*c+1+a*b*c}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{c1-a*c*c3}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{c1-a*c*c3}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{-1}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{-1}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill"}, {"showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "scripts": {}, "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": [{"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{-b*c+a*b*c}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{-b*c+a*b*c}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{a}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{a}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{a*c3-c2}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{a*c3-c2}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{1}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{1}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "0.5", "minValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{z}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "minValue": "{z}", "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill"}, {"showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "scripts": {}, "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        ", "gaps": [{"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{y}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": 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variables", "name": "c"}}, "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-NonCommercial-ShareAlike 4.0 International"}, "name": "Gaussian Elimination", "contributors": [{"name": "Stephen Flood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1397/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Stephen Flood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1397/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}