// Numbas version: exam_results_page_options
{"name": "Solving a Linear and a Non-linear system of equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Solving a Linear and a Non-linear system of equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "statement": "<p>Given two equations:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x+\\var{b1}y=\\var{r1}\\)</p>\n<p style=\"padding-left: 60px;\">and</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}x^2+\\var{d1}y^2=\\var{r2}\\)</p>\n<p>There are two solutions for \\(x\\) that satisfy both of these equations and&nbsp;for each \\(x\\) value there exists a corresponding \\(y\\) value that forms&nbsp;a solution pair.</p>", "preamble": {"js": "", "css": ""}, "functions": {}, "variable_groups": [], "variables": {"a1": {"name": "a1", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "r1": {"name": "r1", "definition": "random(10..20#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "c1": {"name": "c1", "definition": "random(1..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "r2": {"name": "r2", "definition": "random(20..50#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "b1": {"name": "b1", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "d1": {"name": "d1", "definition": "random(2..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}}, "name": "Solving a Linear and a Non-linear system of equations", "variablesTest": {"condition": "(2*{a1}*{d1}*{r1})^2>4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})", "maxRuns": 100}, "advice": "<p>To solve a system that involves&nbsp;a linear equation and a non-linear equation we must use the substitution method.</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x+\\var{b1}y=\\var{r1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}x^2+\\var{d1}y^2=\\var{r2}\\)</p>\n<p>The first equation is a linear equaton, we use this to write one variable in terms of the other.&nbsp;</p>\n<p>For example, we could make y the subject of this equation&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(y=\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\\)</p>\n<p>We can then insert this for every \\(y\\) in the non-linear equation to get</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}x^2+\\var{d1}*\\left(\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\\right)^2=\\var{r2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}x^2+\\var{d1}*\\frac{(\\var{r1}-\\var{a1}x)^2}{\\var{b1}^2}=\\var{r2}\\)</p>\n<p>Multiplying across by \\(\\simplify{{b1}^2}\\) gives</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}*{b1}^2}x^2+\\var{d1}(\\var{r1}-\\var{a1}x)^2-\\simplify{{r2}*{b1}^2}=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}*{b1}^2}x^2+\\var{d1}\\left(\\simplify{{r1}^2}-\\simplify{2*{r1}*{a1}}x+\\simplify{{a1}^2}x^2\\right)-\\simplify{{r2}*{b1}^2}=0\\)</p>\n<p>Gathering the like terms together gives</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{({c1}*{b1}^2+{d1}*{a1}^2)x^2-(2*{a1}*{d1}*{r1})x+({d1}*{r1}^2-{r2}*{b1}^2)}=0\\)</p>\n<p>This is a quadratic equation.</p>\n<p>Recall the formula for solving a quadratic equation of the form &nbsp;\\(ax^2+bx+c=0\\) &nbsp;is given by</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\\)</p>\n<p style=\"padding-left: 30px;\">In this example \\(a =&nbsp;\\simplify{({c1}*{b1}^2+{d1}*{a1}^2)}, b = \\simplify{-(2*{a1}*{d1}*{r1})}\\) and \\(c = \\simplify{{d1}*{r1}^2-{r2}*{b1}^2}\\)</p>\n<p>Once we have each \\(x\\) value we insert it into &nbsp; \\(y=\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\\) &nbsp; to find the corresponding \\(y\\) value.</p>", "parts": [{"showCorrectAnswer": true, "prompt": "<p>Input the larger of the two \\(x\\) values. &nbsp; &nbsp; &nbsp;\\(x = \\)&nbsp;[[0]]</p>\n<p>Input the \\(y\\) value that&nbsp;corresponds to the previous answer. &nbsp; &nbsp; \\(y = \\)&nbsp;[[1]]</p>\n<p></p>\n<p>Input the lesser of the two \\(x\\) values that satisfies both equations. &nbsp; &nbsp; \\(x = \\)&nbsp;[[2]]</p>\n<p>Input the \\(y\\) value that&nbsp;corresponds to the previous answer. &nbsp; &nbsp; \\(y = \\)&nbsp;[[3]]</p>", "variableReplacementStrategy": "originalfirst", "scripts": {}, "showFeedbackIcon": true, "marks": 0, "gaps": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "(2*{a1}*{d1}*{r1}+sqrt{(2*{a1}*{d1}*{r1})^2-4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})})/(2*({b1}^2*{c1}+{d1}*{a1}^2))", "showPrecisionHint": true, "strictPrecision": false, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "minValue": "(2*{a1}*{d1}*{r1}+sqrt{(2*{a1}*{d1}*{r1})^2-4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})})/(2*({b1}^2*{c1}+{d1}*{a1}^2))", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "marks": 1, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionPartialCredit": 0}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "(2*{r1}*{b1}*{c1}-sqrt{(2*{r1}*{b1}*{c1})^2-4*({c1}*{b1}^2+{d1}*{a1}^2)*({c1}*{r1}^2-{r2}*{a1}^2)})/(2*({c1}*{b1}^2+{d1}*{a1}^2))", "showPrecisionHint": true, "strictPrecision": false, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "minValue": "(2*{r1}*{b1}*{c1}-sqrt{(2*{r1}*{b1}*{c1})^2-4*({c1}*{b1}^2+{d1}*{a1}^2)*({c1}*{r1}^2-{r2}*{a1}^2)})/(2*({c1}*{b1}^2+{d1}*{a1}^2))", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "marks": 1, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionPartialCredit": 0}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "(2*{a1}*{d1}*{r1}-sqrt{(2*{a1}*{d1}*{r1})^2-4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})})/(2*({b1}^2*{c1}+{d1}*{a1}^2))", "showPrecisionHint": true, "strictPrecision": false, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "minValue": "(2*{a1}*{d1}*{r1}-sqrt{(2*{a1}*{d1}*{r1})^2-4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})})/(2*({b1}^2*{c1}+{d1}*{a1}^2))", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "marks": 1, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionPartialCredit": 0}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "(2*{r1}*{b1}*{c1}+sqrt{(2*{r1}*{b1}*{c1})^2-4*({c1}*{b1}^2+{d1}*{a1}^2)*({c1}*{r1}^2-{r2}*{a1}^2)})/(2*({c1}*{b1}^2+{d1}*{a1}^2))", "showPrecisionHint": true, "strictPrecision": false, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "minValue": "(2*{r1}*{b1}*{c1}+sqrt{(2*{r1}*{b1}*{c1})^2-4*({c1}*{b1}^2+{d1}*{a1}^2)*({c1}*{r1}^2-{r2}*{a1}^2)})/(2*({c1}*{b1}^2+{d1}*{a1}^2))", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "marks": 1, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionPartialCredit": 0}], "type": "gapfill", "variableReplacements": []}], "extensions": [], "tags": [], "ungrouped_variables": ["a1", "b1", "r1", "c1", "d1", "r2"], "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/"}]}