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{"navigation": {"reverse": true, "onleave": {"action": "none", "message": ""}, "showfrontpage": true, "browse": true, "preventleave": true, "allowregen": true, "showresultspage": "oncompletion"}, "allQuestions": true, "pickQuestions": 0, "type": "exam", "duration": 0, "showQuestionGroupNames": false, "questions": [], "percentPass": 0, "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve a system of linear equations using Gaussian elimination.</p>"}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "pickQuestions": 0, "questions": [{"name": "Gauss elimination to solve a system of linear equations.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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": "George Stagg", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/930/"}], "tags": ["checked2015", "columns", "gauss", "Gauss", "gauss elimination", "Gauss elimination", "Gaussian elimination", "gaussian elimination", "Linear algebra", "Linear Algebra", "linear algebra", "linear equations", "Linear equations", "matrices", "matrix", "reduced matrix", "row operators", "rows", "solving a system of linear equations", "Solving equations", "solving equations"], "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"}, "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>", "advice": "<p>Look at the revealed answers for this question. All the information needed is there.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"x": {"name": "x", "group": "Ungrouped variables", "definition": "c3-(b*a-b)*z-b*y", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "c1+c2*c*(b-a*b)+c3*c*(a^2*b-a-a*b)", "description": "", "templateType": "anything"}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "a*c3-c2-a*z", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}}, "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<p>[[1]]</p>\n<p></p>", "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 1, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "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>"], "matrix": [1, 0, 0, 0], "distractors": ["", "", "", ""]}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "2.4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([\n  [1,b,b*a-b,c3],\n  [a,a*b-1,a^2*b-a-a*b,c2],\n  [a*c,b*c,1,c1]\n])", "correctAnswerFractions": false, "numRows": "3", "numColumns": "4", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "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\" style=\"text-align: center;\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n<td style=\"text-align: center;\">$\\var{1}$</td>\n<td style=\"text-align: center;\">$\\var{b}$</td>\n<td style=\"text-align: center;\">$\\var{b*a-b}$</td>\n<td style=\"text-align: center;\">$\\var{c3}$</td>\n<td rowspan=\"3\" style=\"text-align: center;\">\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\var{0}$</td>\n<td style=\"text-align: center;\">[[2]]</td>\n<td style=\"text-align: center;\">[[3]]</td>\n<td style=\"text-align: center;\">[[4]]</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\var{0}$</td>\n<td style=\"text-align: center;\">[[5]]</td>\n<td style=\"text-align: center;\">[[6]]</td>\n<td style=\"text-align: center;\">[[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": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.8, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{-a}", 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false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.6, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{c2-a*c3}", "maxValue": "{c2-a*c3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.6, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{c*b-c*b*a}", "maxValue": "{c*b-c*b*a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.6, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{-a^2*b*c+1+a*b*c}", "maxValue": "{-a^2*b*c+1+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": 0.6, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{c1-a*c*c3}", "maxValue": "{c1-a*c*c3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.8, "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"}], "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 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\" style=\"text-align: center;\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n<td style=\"text-align: center;\">$\\var{1}$</td>\n<td style=\"text-align: center;\">$\\var{b}$</td>\n<td style=\"text-align: center;\">$\\var{b*a-b}$</td>\n<td style=\"text-align: center;\">$\\var{c3}$</td>\n<td rowspan=\"3\" style=\"text-align: center;\">\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\var{0}$</td>\n<td style=\"text-align: center;\">$\\var{1}$</td>\n<td style=\"text-align: center;\">[[1]]</td>\n<td style=\"text-align: center;\">[[2]]</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\var{0}$</td>\n<td style=\"text-align: center;\">$\\var{0}$</td>\n<td style=\"text-align: center;\">[[3]]</td>\n<td style=\"text-align: center;\">[[4]]</td>\n</tr>\n</tbody>\n</table>\n<p>From this you should find:</p>\n<p>$z=\\;\\;$[[5]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.8, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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", 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"nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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": 0.8, "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": 0.8, "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"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{z}", "maxValue": "{z}", "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": "\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        ", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 3, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{y}", "maxValue": "{y}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": 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