// Numbas version: exam_results_page_options
{"name": "Denis's copy of EG5116 EG4103 Matrices: Cramers Rule applied to 3 simultaneous equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Denis's copy of EG5116 EG4103 Matrices: Cramers Rule applied to 3 simultaneous equations", "tags": [], "metadata": {"description": "<p>Cramers Rule applied to 3 simultaneous equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Using Cramer's rule , solve the system of equations:</p>\n<p>$\\var{a11}x+\\var{a12}y+\\var{a13}z=\\var{c1}$</p>\n<p>$\\var{a21}x+\\var{a22}y+\\var{a23}z=\\var{c2}$</p>\n<p>$\\var{a31}x+\\var{a32}y+\\var{a33}z=\\var{c3}$</p>\n<p></p>\n<p></p>", "advice": "<p>If \\[&nbsp; A=\\left( \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; a_{13} \\\\a_{21} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right),\\]</p>\n<p>\\[&nbsp; C=\\left( \\begin{array}{ccc}<br/>c_{1} \\\\&nbsp;c_{2}&nbsp;\\\\c_{3} \\end{array} \\right),\\]</p>\n<p>Cramer's Rule : ${x_1}=\\frac{\\Delta_1}{\\Delta_0}$ ,&nbsp; ${x_2}=\\frac{\\Delta_2}{\\Delta_0}$ , ${x_3}=\\frac{\\Delta_3}{\\Delta_0}$</p>\n<p>Where:\\[ \\Delta_0=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; a_{13} \\\\a_{21} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_1=\\left| \\begin{array}{ccc}<br/>c_{1} &amp;&nbsp;a_{12} &amp; a_{13} \\\\c_{2} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ c_{3} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_2=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;c_{1} &amp; a_{13} \\\\a_{21} &amp;&nbsp;c_{2} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;c_{3} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_3=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; c_{1} \\\\a_{21} &amp;&nbsp;a_{22} &amp; c_{2}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; c_{3}\\end{array} \\right|\\]</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p><span class=\"nb\"><span class=\"k\"><span class=\"s\">&nbsp;</span></span></span></p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"a33": {"name": "a33", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(0..10 except(a21*a12/a11))", "description": "", "templateType": "anything"}, "a23": {"name": "a23", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a13": {"name": "a13", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a31": {"name": "a31", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "matrixA": {"name": "matrixA", "group": "Ungrouped variables", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "description": "", "templateType": "anything"}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "a31*x1+a32*x2+a33*x3", "description": "", "templateType": "anything"}, "matrixA3": {"name": "matrixA3", "group": "Cramer determinants", "definition": "matrix([a11,a12,c1],[a21,a22,c2],[a31,a32,c3])", "description": "", "templateType": "anything"}, "matrixA2": {"name": "matrixA2", "group": "Cramer determinants", "definition": "matrix([a11,c1,a13],[a21,c2,a23],[a31,c3,a33])", "description": "", "templateType": "anything"}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "a21*x1+a22*x2+a23*x3", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "a11*x1+a12*x2+a13*x3", "description": "", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "matrixA1": {"name": "matrixA1", "group": "Cramer determinants", "definition": "matrix([c1,a12,a13],[c2,a22,a23],[c3,a32,a33])", "description": "", "templateType": "anything"}, "a32": {"name": "a32", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "x1", "x2", "x3", "c1", "c2", "c3"], "variable_groups": [{"name": "Cramer determinants", "variables": ["matrixA1", "matrixA2", "matrixA3"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>What is the determinant of A=$\\var{matrixA}$? i.e $\\Delta$</p>\n<p>[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(matrixA)", "maxValue": "det(matrixA)", "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, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate&nbsp;$\\Delta_x$ [[0]]</p>\n<p>Hence, calculate ${x}$&nbsp; [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(matrixA1)", "maxValue": "det(matrixA1)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{x1}", "maxValue": "{x1}", "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, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate&nbsp;$\\Delta_y$[[0]]</p>\n<p>Hence, calculate ${y}$&nbsp; [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(matrixA2)", "maxValue": "det(matrixA2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{x2}", "maxValue": "{x2}", "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, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate&nbsp;$\\Delta_z$[[0]]</p>\n<p>Hence, calculate ${z}$&nbsp; [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(matrixA3)", "maxValue": "det(matrixA3)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{x3}", "maxValue": "{x3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}]}]}], "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}]}