// Numbas version: exam_results_page_options
{"name": "Simultaneous equations: linear and quadratic, 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": ["roots", "root1", "root2", "grad", "sroots", "proots", "yint", "quadxcoeff", "quadccoeff", "ansy1", "ansy2"], "name": "Simultaneous equations: linear and quadratic, two points", "tags": ["algebra", "equations", "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;\">$\\simplify{x^2+{quadxcoeff}x+{quadccoeff}}$</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": false, "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": false, "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>There are many ways to solve these equations simultaneously. Here is one method.</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;\">$\\simplify{x^2+{quadxcoeff}x+{quadccoeff}}$</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} =x^2+{quadxcoeff}x+{quadccoeff}}\\]</p>\n<p>Since we have a quadratic here we get everything onto one side:<br/>\\[0=\\simplify{x^2+{sroots}x+{proots}}\\]</p>\n<p>There are various ways to solve a quadratic, in this particular case we can factorise the quadratic:</p>\n<p>\\[(\\simplify{x-{root1}})(\\simplify{x-{root2}})=0\\]</p>\n<p>Therefore,&nbsp;$x=\\var{root1},\\,\\var{root2}$.</p>\n<p><br/>Now for&nbsp;$x=\\var{root1}$, we can determine the corresponding $y$ value by substituting $x=\\var{root1}$ 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;\">$\\simplify[!collectnumbers]{{grad}({root1})+{yint}}$</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;$x=\\var{root2}$, so we can determine the corresponding $y$ value by substituting $x=\\var{root2}$ 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;\">$\\simplify[!collectnumbers]{{grad}({root2})+{yint}}$</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 $x_1=\\var{root1}$,&nbsp;$y_1=\\var{ansy1}$ and&nbsp;$x_2=\\var{root2}$,&nbsp;$y_2=\\var{ansy2}$.</p>\n<p>In other words, the two curves intersect at the points $(\\var{root1},\\var{ansy1})$ and&nbsp;$(\\var{root2},\\var{ansy2})$.</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": {"quadxcoeff": {"definition": "sroots+grad", "templateType": "anything", "group": "Ungrouped variables", "name": "quadxcoeff", "description": ""}, "yint": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "yint", "description": ""}, "root1": {"definition": "roots[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "root1", "description": ""}, "proots": {"definition": "root1*root2", "templateType": "anything", "group": "Ungrouped variables", "name": "proots", "description": ""}, "quadccoeff": {"definition": "proots+yint", "templateType": "anything", "group": "Ungrouped variables", "name": "quadccoeff", "description": ""}, "ansy2": {"definition": "grad*root2+yint", "templateType": "anything", "group": "Ungrouped variables", "name": "ansy2", "description": ""}, "sroots": {"definition": "-root1-root2", "templateType": "anything", "group": "Ungrouped variables", "name": "sroots", "description": ""}, "ansy1": {"definition": "grad*root1+yint", "templateType": "anything", "group": "Ungrouped variables", "name": "ansy1", "description": ""}, "grad": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "grad", "description": ""}, "roots": {"definition": "sort(shuffle(-12..12)[0..2])", "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": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}