// Numbas version: exam_results_page_options
{"name": "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": [{"name": "Gauss elimination to solve a system of linear equations", "variable_groups": [], "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"}, "ungrouped_variables": ["a", "c", "b", "c3", "c2", "y", "x", "c1", "z"], "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "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>", "preamble": {"js": "", "css": ""}, "tags": [], "parts": [{"customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "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        ", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "gaps": [{"customMarkingAlgorithm": "", "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>"], "maxMarks": 1, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "shuffleChoices": true, "showFeedbackIcon": true, "marks": 0, 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"{c1}"}], "variableReplacements": []}, {"customMarkingAlgorithm": "", "sortAnswers": false, "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>$\\var{c3}$</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 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"marks": 0.6, "unitTests": [], "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "minValue": "{c1-a*c*c3}"}, {"customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{-1}", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 0.8, "unitTests": [], "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "minValue": "{-1}"}], "variableReplacements": []}, {"customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "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        ", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "gaps": [{"customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{-b*c+a*b*c}", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 0.8, "unitTests": [], "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "minValue": "{-b*c+a*b*c}"}, {"customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{a}", "mustBeReducedPC": 0, 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"extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 0.8, "unitTests": [], "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "minValue": "{1}"}, {"customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 0.8, "unitTests": [], "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "minValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}"}, {"customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{z}", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 2, "unitTests": [], "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "minValue": "{z}"}], "variableReplacements": []}, {"customMarkingAlgorithm": "", "sortAnswers": false, "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        ", 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