// Numbas version: finer_feedback_settings
{"name": "Solving Simultaneous Equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solving Simultaneous Equations", "tags": [], "metadata": {"description": "<p>Semi-worked example of solving simultaneous equations using matrices.&nbsp;Equation values are randomly generated. The student is walked through the steps needed to solve the equations.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h2>Solving Simultaneous Equations using Matrices</h2>\n<p></p>\n<p>Simultaneous equations consist of two equations with two unknown values and&nbsp;both equations must be solved at the same time. This can be done using matrices.</p>\n<p></p>\n<p>Consider the following set of simultaneous equations:</p>\n<p>\\begin{align}<br/>\\simplify{{mA[0][0]}x + {mA[0][1]}y} &amp;=&nbsp;\\var{mC[0][0]} \\\\<br/>\\simplify{{mA[1][0]}x + {mA[1][1]}y} &amp;=&nbsp;\\var{mC[1][0]}<br/>\\end{align}</p>\n<p>We can solve these equations using the following steps:</p>\n<p>(<em>Input all numbers as fractions or integers and not as decimals.)</em></p>", "advice": "", "rulesets": {}, "extensions": [], "variables": {"x": {"name": "x", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,1,1,-1)", "description": "<p>value for x</p>", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,1,1,-1)", "description": "<p>value for y</p>", "templateType": "anything"}, "mA": {"name": "mA", "group": "Ungrouped variables", "definition": "matrixRandomNInvertable(matrix(repeat(repeat(random(1..9)*random(1,1,1,-1),2),2)))", "description": "", "templateType": "anything"}, "mC": {"name": "mC", "group": "Ungrouped variables", "definition": "matrix([ma[0][0]*x+ma[0][1]*y],[ma[1][0]*x+ma[1][1]*y])", "description": "", "templateType": "anything"}, "mAI": {"name": "mAI", "group": "Ungrouped variables", "definition": "matrix([mA[1][1],-mA[0][1]],[-mA[1][0],ma[0][0]])/det(mA)", "description": "", "templateType": "anything"}, "det_mA": {"name": "det_mA", "group": "Ungrouped variables", "definition": "det(mA)", "description": "", "templateType": "anything"}, "sx": {"name": "sx", "group": "Ungrouped variables", "definition": "'x'", "description": "", "templateType": "anything"}, "sy": {"name": "sy", "group": "Ungrouped variables", "definition": "'y'", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "det_mA<>0", "maxRuns": 100}, "ungrouped_variables": ["x", "y", "mA", "mC", "mAI", "det_mA", "sx", "sy"], "variable_groups": [], "functions": {"matrixRandomNInvertable": {"parameters": [["ma", "matrix"]], "type": "matrix", "language": "jme", "definition": "if(det(ma)<>0,ma,\n  matrixRandomNInvertable(\n    matrix(repeat(repeat(random(1..9)*(random(1,1,1,-1)),numcolumns(ma)),numrows(ma)))\n  )\n)"}}, "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>First, rewrite the equations in matrix form of $Ab = C$, where the matrix $A$ will contain the coefficients of the equation, $b$ will contain the unknown values and $C$ will contain the constants.</p>\n<div class=\"matrix-input\" style=\"text-align: left;\">\n<div class=\"matrix-wrapper\"></div>\n<div class=\"matrix-wrapper\"><span>$\\mathbf{A} = $&nbsp;[[0]]</span>&nbsp; &nbsp; $\\mathbf{b} = $&nbsp;\n<div class=\"matrix-input\">\n<div class=\"matrix-wrapper\"><span class=\"left-bracket\"></span>\n<table class=\"matrix\">\n<tbody>\n<tr>\n<td>[[1]]</td>\n</tr>\n<tr>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>\n<span class=\"right-bracket\"></span></div>\n</div>\n&nbsp; &nbsp; &nbsp; &nbsp;$\\mathbf{C} = $&nbsp;[[3]]</div>\n</div>", "stepsPenalty": 0, "steps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The coefficients are the values $x$ and $y$ are multiplied by. Take the number of $x$s in the first equation and place it in the first element of matrix $A$, $a_{11}$.&nbsp;Take the number of $y$s in the first equation and place it in the second element of matrix $A$, $a_{12}$ (first row, second column).</p>\n<p>Repeat for the second equation, but place coefficients into the second row elements of matrix $A$.</p>", "correctAnswer": "mA", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": "0.003", "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "information", "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>The unknown values $x$ and $y$ go into matrix $b$, $x$ into $b_{11}$ and $y$ into&nbsp;$b_{21}$. We don't know these values yet, so we use&nbsp;$x$ and $y$ for now.</p>"}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The constants are the answers to the equations (the number to the right of the $=$ sign). The answer to the first equation goes into the first element of matrix $C$, $c_{11}$.&nbsp;The answer to the second equation goes into the&nbsp;second element of matrix $C$, $c_{21}$.</p>", "correctAnswer": "mC", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": "0.003", "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "mA", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "y", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "y", "value": ""}]}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "mC", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Second, we find the inverse of the coefficient&nbsp;matrix $A$</p>", "stepsPenalty": 0, "steps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Switch the elements on the leading/main diagonal.</p>\n<p>Change the sign of the elelments on the secondary diagonal.</p>\n<p>i.e. $\\begin{pmatrix} d &amp; -b\\\\ -c &amp; a \\end{pmatrix}$</p>", "correctAnswer": "matrix([mA[1][1],-mA[0][1]],[-mA[1][0],ma[0][0]])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the determinant of $A$</p>\n<p>i.e. $ad-bc$</p>", "minValue": "det(mA)", "maxValue": "det(mA)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Divide every element in $A'$ by the determinant. <em>(Enter answer as fractions.)</em></p>", "correctAnswer": "mAI", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": "0.003", "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "correctAnswer": "mAI", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": "0.003", "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Finally, find $A^{-1}C$. i.e&nbsp;multiply the inverse of $A$, $A^{-1}$, by $C$</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>We have matrix $A^{-1} = \\begin{pmatrix} a_{11} &amp; a_{12}\\\\ a_{21} &amp; a_{22} \\end{pmatrix}$ multiplied by matrix $C = \\begin{pmatrix} c_{11}\\\\ c_{21} \\end{pmatrix}$.</p>\n<p>We know we can multiply these as the number of columns in $A^{-1}$, 2, is equal to the number of rows in $C$, 2.</p>\n<p>[2 x 2] x [2 x 1]</p>\n<p>We can also determine the order of the answer matrix:-</p>\n<ul>\n<li>There are 2 rows in matrix $A^{-1}$</li>\n<li>There are 1 columns in matrix $C$</li>\n<li>Therefore the answer will be a 2 x 1 matrix, $A^{-1}C = \\begin{pmatrix} a'c_{11}\\\\ a'c_{21} \\end{pmatrix}$</li>\n</ul>\n<p>To work out the values for the answer matrix we multiply:-</p>\n<ul>\n<li>Each value in row 1 in matrix $A^{-1}$ by the corresponding value in column 1 in matrix $C$ &ndash; and work out the total. This provides the value for $a'c_{11}$</li>\n<li>Each value in row 2 in matrix $A^{-1}$ by the corresponding value in column 1 in matrix $C$ &ndash; and work out the total. This provides the value for&nbsp;$a'c_{21}$</li>\n</ul>"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$a'c_{11} = (a'_{11}\\times c_{11}) + (a'_{12}\\times c_{21}) =$ ?</p>", "minValue": "x", "maxValue": "x", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$a'c_{21} = (a'_{21}\\times c_{11}) + (a'_{22}\\times c_{21}) =$ ?</p>", "minValue": "y", "maxValue": "y", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Therefore $A^{-1}C = \\begin{pmatrix} a'c_{11}\\\\ a'c_{21} \\end{pmatrix}$ is?</p>", "correctAnswer": "matrix([x],[y])", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": "0.003", "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "correctAnswer": "matrix([x],[y])", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": "0.003", "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"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>We started with the matrices $Ab=C$, and we have rearranged this formula to $A^{-1}C=b$ in order to find $b$. Therefore, $x$ and $y$ are?</p>\n<p>$x = $ [[0]]</p>\n<p>$y = $ [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "x", "maxValue": "x", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "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"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Mark Patterson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5064/"}]}]}], "contributors": [{"name": "Mark Patterson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5064/"}]}