// Numbas version: finer_feedback_settings
{"name": "Solving two linear equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "<p>Input the value for \\(x\\)&nbsp;as an exact fraction.</p>\n<p style=\"padding-left: 30px;\">\\(x = \\)&nbsp;[[0]]</p>\n<p>Input the value for \\(y\\)&nbsp;as an exact fraction.</p>\n<p style=\"padding-left: 30px;\">\\(y = \\)&nbsp;[[1]]</p>", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "({d}*{r1}-{b}*{r2})/({a}*{d}-{b}*{c})", "mustBeReduced": false, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": true, "allowFractions": true, "minValue": "({d}*{r1}-{b}*{r2})/({a}*{d}-{b}*{c})", "marks": 1, "variableReplacements": [], "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "(-{c}*{r1}+{a}*{r2})/({a}*{d}-{b}*{c})", "mustBeReduced": false, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": true, "allowFractions": true, "minValue": "(-{c}*{r1}+{a}*{r2})/({a}*{d}-{b}*{c})", "marks": 1, "variableReplacements": [], "showFeedbackIcon": true}]}], "statement": "<p>Solve the following system of simultaneous equations:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a}x+\\var{b}y=\\var{r1}\\)</p>\n<p style=\"padding-left: 60px;\">and</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c}x+\\var{d}y=\\var{r2}\\)</p>", "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a", "b", "c", "d", "r1", "r2"], "name": "Solving two linear equations", "functions": {}, "extensions": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solving two simultaneous linear equations</p>"}, "variablesTest": {"condition": "{a}*{d}>{b}*{c}\n", "maxRuns": 100}, "advice": "<p>equation (i) &nbsp; &nbsp;\\(\\var{a}x+\\var{b}y=\\var{r1}\\)</p>\n<p>equation (ii) &nbsp; &nbsp;\\(\\var{c}x+\\var{d}y=\\var{r2}\\)</p>\n<p>If we decide to eliminate the \\(x\\) variables we need to have the same number of \\(x\\) in both equations</p>\n<p>\\(\\var{c}\\)*equation (i) &nbsp; &nbsp; &nbsp;\\(\\simplify{{c}*{a}}x+\\simplify{{c}*{b}}y=\\simplify{{c}*{r1}}\\)</p>\n<p>\\(\\var{a}\\)*equation (ii) &nbsp; &nbsp;&nbsp;\\(\\simplify{{c}*{a}}x+\\simplify{{d}*{a}}y=\\simplify{{a}*{r2}}\\)</p>\n<p>Subtracting gives:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(\\simplify{{c}*{b}-{d}*{a}}y=\\simplify{{c}*{r1}-{a}*{r2}}\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(y=\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})}\\)</p>\n<p>Substituting this solution for \\(y\\) into equation (i) gives</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp;\\(\\var{a}x+\\var{b}*(\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})})=\\var{r1}\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\\(\\var{a}x=\\var{r1}-\\var{b}*(\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})})\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\\(\\var{a}x=\\simplify{{r1}-{b}*({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})}\\)</p>\n<p></p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\\(x=\\simplify{({r1}-{b}*({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a}))/{a}}\\)</p>\n<p></p>", "preamble": {"css": "", "js": ""}, "variables": {"b": {"name": "b", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..12#1)"}, "d": {"name": "d", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..12#1)"}, "r1": {"name": "r1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(10..40#1)"}, "c": {"name": "c", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..11#1)"}, "a": {"name": "a", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..10#1)"}, "r2": {"name": "r2", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(16..50#1)"}}, "tags": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}