// Numbas version: exam_results_page_options
{"name": "Gaussian 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": "Gaussian elimination to solve a system of linear equations", "tags": [], "metadata": {"description": "<p>Solving a system of three linear equations in 3 unknowns using Gaussian Elimination (or Gauss-Jordan algorithm) in 5 stages. Solutions are all integers. Introductory question where the numbers come out quite nice with not much dividing. Set-up is meant for formative assessment. Adapated from a question copied from Newcastle. </p>", "licence": "Creative Commons Attribution 4.0 International"}, "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 column below the second entry in the second row using the second row. <br/><strong>Part d)&nbsp;</strong>Introduce zeros in the third column above the third entry in the third row using the third row. <br/><strong>Part e)</strong> Introduce zeros in the second column above the second entry in the second row using the second row.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Also read off the solution to the system.</p>", "advice": "<p>Look at the revealed answers for this question. All the information needed is there.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"x": {"name": "x", "group": "Ungrouped variables", "definition": "c3-(b*a-b)*z-b*y", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "c1+c2*c*(b-a*b)+c3*c*(a^2*b-a-a*b)", "description": "", "templateType": "anything", "can_override": false}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "a*c3-c2-a*z", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "c3", "c2", "y", "x", "c1", "z"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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{.}\\\\ \\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{.}\\\\ \\phantom{.}\\end{matrix} \\middle| 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"allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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{.}\\\\\\phantom{.} \\end{matrix} 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"sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<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 (which is the leading \\(1\\) of the second row) 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{.}\\\\ \\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{.}\\\\ \\phantom{.}\\end{matrix} \\middle| \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\right.\\]</td>\n<td>$\\var{c3}$</td>\n<td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\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></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1.75", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": true, "customName": "negative", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>You seem to have got the sign wrong.</p>", "useAlternativeFeedback": false, "minValue": "{b*c-a*b*c}", "maxValue": "{b*c-a*b*c}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "{-b*c+a*b*c}", "maxValue": "{-b*c+a*b*c}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.375", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": true, "customName": "negative", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>It looks like you forgot to multiply the second row going from the previous matrix to this one.</p>", "useAlternativeFeedback": false, "minValue": "{-a}", "maxValue": "{-a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "{a}", "maxValue": "{a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.375", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": true, "customName": "negative", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>It looks like you forgot to multiply the second row going from the previous matrix to this one.</p>", "useAlternativeFeedback": false, "minValue": "{-a*c3+c2}", "maxValue": "{-a*c3+c2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "{a*c3-c2}", "maxValue": "{a*c3-c2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.75", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{1}", "maxValue": "{1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.75", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "maxValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>To read off the solutions to \\(x\\), \\(y\\) and \\(z\\) most easily, we now want to get the matrix into reduced echelon form.</p>\n<p>We first introduce $0$s in the third column above the leading one in the third row by adding:<br/>[[0]] times the third row to the second row and<br/>[[1]] times the third row to the first row to get the matrix:</p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\right.\\]</td>\n<td>$\\var{1}$</td>\n<td>$\\var{b}$</td>\n<td>$\\var{0}$</td>\n<td rowspan=\"3\">\\[\\left. \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\middle| \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\right.\\]</td>\n<td>[[2]]</td>\n<td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\right.\\]</td>\n</tr>\n<tr>\n<td>$\\var{0}$</td>\n<td>$\\var{1}$</td>\n<td>\\(\\var{0}\\)</td>\n<td>[[3]]</td>\n</tr>\n<tr>\n<td>$\\var{0}$</td>\n<td>$\\var{0}$</td>\n<td>\\(\\var{1}\\)</td>\n<td>[[4]]</td>\n</tr>\n</tbody>\n</table>\n<p>Finally, we introduce a $0$ in the second column above the leading \\(1\\) in the second row by adding:<br/>[[5]] times the second row to the first row to get the matrix:</p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\right.\\]</td>\n<td>$\\var{1}$</td>\n<td>\\(\\var{0}\\)</td>\n<td>$\\var{0}$</td>\n<td rowspan=\"3\">\\[\\left. \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\middle| \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\right.\\]</td>\n<td>[[6]]</td>\n<td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\end{matrix} \\right.\\]</td>\n</tr>\n<tr>\n<td>$\\var{0}$</td>\n<td>$\\var{1}$</td>\n<td>\\(\\var{0}\\)</td>\n<td>[[3]]</td>\n</tr>\n<tr>\n<td>$\\var{0}$</td>\n<td>$\\var{0}$</td>\n<td>\\(\\var{1}\\)</td>\n<td>[[4]]</td>\n</tr>\n</tbody>\n</table>\n<p>From this you can read off the solution (the gaps will fill automatically from your answers in the augmented matrix):</p>\n<p>\\(x=\\ \\) [[6]]</p>\n<p>\\(y=\\ \\) [[3]]</p>\n<p>\\(z=\\ \\) [[4]]</p>", "gaps": 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"{-a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1.75", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": true, "customName": "negative", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>You seem to have got the sign wrong.</p>", "useAlternativeFeedback": false, "minValue": "{-b*(1-a)}", "maxValue": "{-b*(1-a)}", 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