// Numbas version: finer_feedback_settings
{"name": "Third Form 2018 Question 15 (Essay)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "{a}<>{a1} and {b}<>{b1}", "maxRuns": 100}, "variables": {"a1": {"group": "Ungrouped variables", "definition": "random(1..9)", "name": "a1", "description": "", "templateType": "anything"}, "x": {"group": "Ungrouped variables", "definition": "random(-20..20)", "name": "x", "description": "", "templateType": "anything"}, "y": {"group": "Ungrouped variables", "definition": "random (-20..20 except x)", "name": "y", "description": "", "templateType": "anything"}, "b1": {"group": "Ungrouped variables", "definition": "random(1..9 except round(a1*b/a))", "name": "b1", "description": "", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "definition": "random(1..9)", "name": "a", "description": "", "templateType": "anything"}, "c1": {"group": "Ungrouped variables", "definition": "a1*x+b1*y", "name": "c1", "description": "", "templateType": "anything"}, "c": {"group": "Ungrouped variables", "definition": "a*x+b*y", "name": "c", "description": "", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "definition": "random(-9..9 except [0,a])", "name": "b", "description": "", "templateType": "anything"}}, "tags": [], "statement": "<p>\\[\\begin{eqnarray*} \\simplify{{a}x+{b}y}&amp;=&amp;\\var{c}\\\\\\\\\\simplify{{a1}x+{b1}y}&amp;=&amp;\\var{c1}\\end{eqnarray*}\\]</p>", "name": "Third Form 2018 Question 15 (Essay)", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Shows how to define variables to stop degenerate examples.</p>"}, "advice": "<p>Multiply the first equation by $\\var{b1}$ and the second equation by $\\var{b}$ so they both have the same $y$ coefficient:</p>\n<p>\\begin{align}<br/>\\simplify{{a*b1}x+{b*b1}y} &amp;= \\var{c*b1} \\\\<br/>\\simplify{{a1*b}x+{b1*b}y} &amp;= \\var{c1*b}<br/>\\end{align}</p>\n<p>Next, subtract the second equation from the first to get</p>\n<p>\\[ \\simplify[std]{{a*b1-a1*b}x} = \\var{c*b1-c1*b} \\]</p>\n<p>So $x = \\simplify[std]{{(c*b1-c1*b)/(a*b1-a1*b)}}$.</p>\n<p>Substitute this value of $x$ into the first equation and rearrange to obtain $y$:</p>\n<p>\\begin{align}<br/>\\simplify[std]{{a}*{(c*b1-c1*b)/(a*b1-a1*b)} + {b}y} &amp;= \\var{c} \\\\<br/>\\simplify[std]{{b}y} &amp;= \\simplify[std]{{c}-{a*(c*b1-c1*b)/(a*b1-a1*b)}} \\\\<br/>y &amp;= \\simplify[std]{{(c-a*(c*b1-c1*b)/(a*b1-a1*b))/b}}<br/>\\end{align}</p>", "functions": {}, "rulesets": {"std": ["All", "fractionnumbers"]}, "ungrouped_variables": ["a", "c", "a1", "b1", "b", "c1", "x", "y"], "preamble": {"js": "", "css": ""}, "extensions": [], "parts": [{"type": "gapfill", "gaps": [{"answer": "{b1*c-b*c1}/{a*b1-a1*b}", "type": "jme", "vsetRange": [0, 1], "scripts": {}, "unitTests": [], "answerSimplification": "Std", "showCorrectAnswer": true, "showPreview": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "expectedVariableNames": [], "marks": 1, "variableReplacements": [], "checkVariableNames": false, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "failureRate": 1, "notallowed": {"partialCredit": 0, "showStrings": false, "strings": ["."], "message": "<p>Input your answer as a fraction and not a decimal.</p>"}}, {"answer": "{-c*a1+c1*a}/{a*b1-a1*b}", "type": "jme", "vsetRange": [0, 1], "scripts": {}, "unitTests": [], "answerSimplification": "Std", "showCorrectAnswer": true, "showPreview": true, "vsetRangePoints": 5, "showFeedbackIcon": true, "expectedVariableNames": [], "marks": 1, "variableReplacements": [], "checkVariableNames": false, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "failureRate": 1, "notallowed": {"partialCredit": 0, "showStrings": false, "strings": ["."], "message": "<p>Input your answer as a fraction and not as a decimal.</p>"}}], "scripts": {}, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "variableReplacements": [], "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "<p>$x=$&nbsp;[[0]]</p>\n<p>$y=$&nbsp;[[1]]</p>\n<p>Input your answers as fractions and not as decimals.</p>"}], "variable_groups": [], "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}, {"name": "sean hunte", "profile_url": 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