// Numbas version: exam_results_page_options
{"name": "Rob's copy of Simultaneous equations: linear and hyperbola, two points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "grad", "yint", "num", "quadxcoeff", "quadccoeff", "ansy1", "ansy2", "roots", "root1", "root2"], "name": "Rob's copy of Simultaneous equations: linear and hyperbola, two points", "tags": ["algebra", "equations", "hyperbola", "quadratic", "Simultaneous equations", "simultaneous equations", "solving equations", "Solving equations", "system of equations"], "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "2", "prompt": "<p>Find the $x$ and $y$ values that satisfy both of the following equations. That is, find the points of intersection of the two curves.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{grad}x+{yint}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(1)$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{num}/x+{d}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(2)$</td>\n</tr>\n</tbody>\n</table>\n<p>$x_1=$&nbsp;[[0]], &nbsp; $y_1=$&nbsp;[[1]] and $x_2=$ [[2]], $y_2=$&nbsp;[[3]]</p>\n<p><strong>Note:</strong>&nbsp;To input your answer<span>&nbsp;please ensure that $x_1&lt;x_2$.</span></p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "root1", "minValue": "root1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": true, "variableReplacements": [], "maxValue": "ansy1", "minValue": "ansy1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": true, "variableReplacements": [], "maxValue": "root2", "minValue": "root2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": true, "variableReplacements": [], "maxValue": "ansy2", "minValue": "ansy2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>Given</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{grad}x+{yint}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(1)$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{num}/x+{d}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(2)$</td>\n</tr>\n</tbody>\n</table>\n<p>substitute the expression for $y$ given in $(1)$ into $(2)$:<br/>\\[\\simplify{{grad}x+{yint} ={num}/x+{d}}\\]</p>\n<p>To get rid of the&nbsp;$x$ in the denominator, let us multiply both sides by $x$<br/>\\[\\simplify{{grad}x^2+{yint}x ={num}+{d}x}\\]</p>\n<p>Notice this equation is a quadratic, we put everything on one side</p>\n<p>\\[\\simplify{{grad}x^2+{yint-d}x -{num}=0}\\]</p>\n<p>There are various ways to solve a quadratic, in this particular case we can factorise the quadratic:</p>\n<p>\\[(\\simplify{{a}x+{b}})(\\simplify{x+{c}})=0\\]</p>\n<p>Therefore,&nbsp;$x=\\simplify{{-b}/{a}},\\,\\var{-c}$.</p>\n<p><br/>Now for&nbsp;$x=\\simplify[fractionnumbers]{{root1}}$, we can determine the corresponding $y$ value by substituting $x=\\simplify[fractionnumbers]{{root1}}$ into either equation $(1)$ or $(2)$. Below we substitute into $(1)$:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!collectnumbers,fractionnumbers]{{grad}({root1})+{yint}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{ansy1}$</td>\n</tr>\n</tbody>\n</table>\n<p>Now for&nbsp;$x=\\simplify[fractionnumbers]{{root2}}$, so we can determine the corresponding $y$ value by substituting $x=\\simplify[fractionnumbers]{{root2}}$ into either equation $(1)$ or $(2)$.&nbsp;Below we substitute into $(1)$:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!collectnumbers,fractionnumbers]{{grad}({root2})+{yint}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{ansy2}$</td>\n</tr>\n</tbody>\n</table>\n<p>Therefore the values that satisfy equations $(1)$ and $(2)$ are $x_1=\\simplify[fractionnumbers]{{root1}}$,&nbsp;$y_1=\\var{ansy1}$ and&nbsp;$x_2=\\simplify[fractionnumbers]{{root1}}$,&nbsp;$y_2=\\var{ansy2}$.</p>\n<p>In other words, the two curves intersect at the points $\\left(\\simplify[fractionnumbers]{{root1}},\\var{ansy1}\\right)$ and&nbsp;$\\left(\\simplify[fractionnumbers]{{root2}},\\var{ansy2}\\right)$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2,3,4,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-12..12 except [b/a,0])", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "quadxcoeff": {"definition": "yint-d", "templateType": "anything", "group": "Ungrouped variables", "name": "quadxcoeff", "description": ""}, "d": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "yint": {"definition": "a*c+b+d", "templateType": "anything", "group": "Ungrouped variables", "name": "yint", "description": ""}, "root1": {"definition": "roots[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "root1", "description": ""}, "quadccoeff": {"definition": "-num", "templateType": "anything", "group": "Ungrouped variables", "name": "quadccoeff", "description": ""}, "num": {"definition": "-b*c", "templateType": "anything", "group": "Ungrouped variables", "name": "num", "description": ""}, "ansy2": {"definition": "grad*(root2)+yint", "templateType": "anything", "group": "Ungrouped variables", "name": "ansy2", "description": ""}, "ansy1": {"definition": "grad*(root1)+yint", "templateType": "anything", "group": "Ungrouped variables", "name": "ansy1", "description": ""}, "grad": {"definition": "a", "templateType": "anything", "group": "Ungrouped variables", "name": "grad", "description": ""}, "roots": {"definition": "sort([-b/a,-c])", "templateType": "anything", "group": "Ungrouped variables", "name": "roots", "description": ""}, "root2": {"definition": "roots[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "root2", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Rob Cade", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/70/"}]}]}], "contributors": [{"name": "Rob Cade", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/70/"}]}