// Numbas version: exam_results_page_options
{"name": "Simultaneous equations: simple quadratic and hyperbola, one point", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "preamble": {"js": "", "css": ""}, "tags": ["algebra", "equations", "hyperbola", "quadratic", "Simultaneous equations", "simultaneous equations", "solving equations", "Solving equations", "system of equations"], "ungrouped_variables": ["xans", "cubed", "a", "ak", "b", "yans"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0, "name": ""}], "advice": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"scripts": {}, "variableReplacements": [], "prompt": "<p>Find the $x$ and $y$ values that satisfy both of the following equations. That is, find the point 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;\">$\\simplify{{ak}/({b}x)}$</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;\">$\\simplify{{a}x^2/{b}}$</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>$x=$&nbsp;[[0]], &nbsp; $y=$&nbsp;[[1]]</p>\n<p></p>", "stepsPenalty": 0, "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "showCorrectAnswer": true, "minValue": "xans", "scripts": {}, "showPrecisionHint": false, "maxValue": "xans"}, {"allowFractions": true, "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showCorrectAnswer": true, "minValue": "a*xans^2/b", "scripts": {}, "showPrecisionHint": false, "maxValue": "a*xans^2/b"}], "showCorrectAnswer": true, "marks": 0, "type": "gapfill", "steps": [{"scripts": {}, "variableReplacements": [], "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;\">$\\simplify{{ak}/({b}x)}$</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;\">$\\simplify{{a}x^2/{b}}$</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{{ak}/({b}x) ={a}x^2/{b}}\\]</p>\n<p>To get rid of the&nbsp;$x$ in the denominator, let us multiply both sides by $x$<br/> \\[\\simplify{{ak}/({b}) ={a}x^3/{b}}\\]</p>\n<p>Since there is only one term with an $x$ in it, we can get $x^3$ by itself</p>\n<p>\\[x^3=\\var{cubed}\\]</p>\n<p>Therefore,&nbsp;$x=\\sqrt[3]{\\var{cubed}}=\\var{xans}$.</p>\n<p><br/> Now we know the $x$ value we can determine the corresponding $y$ value by substituting $x=\\var{xans}$ into either equation $(1)$ or $(2)$, below we substitute into $(2)$:</p>\n<table>\n<tbody>\n<tr>\n<td>\n<p>$y$</p>\n</td>\n<td>\n<p>$=$</p>\n</td>\n<td style=\"text-align: left;\">\n<p>$\\simplify{{a}/{b}}(\\var{xans})^2$</p>\n</td>\n</tr>\n<tr>\n<td></td>\n<td>\n<p>$=$</p>\n</td>\n<td style=\"text-align: left;\">\n<p>$\\simplify{{a*xans^2/b}}$</p>\n</td>\n</tr>\n</tbody>\n</table>\n<p>Therefore the values that satisfy equations $(1)$ and $(2)$ are $x=\\var{xans}$ and $y=\\simplify{{a*xans^2/b}}$.</p>\n<p>&nbsp;</p>", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "type": "information"}]}], "showQuestionGroupNames": false, "statement": "", "variables": {"yans": {"description": "", "group": "Ungrouped variables", "name": "yans", "definition": "a*cubed/b", "templateType": "anything"}, "a": {"description": "", "group": "Ungrouped variables", "name": "a", "definition": "random(-4..4 except 0)", "templateType": "anything"}, "b": {"description": "", "group": "Ungrouped variables", "name": "b", "definition": "random(-12..12 except [a,0,1,-1])", "templateType": "anything"}, "ak": {"description": "", "group": "Ungrouped variables", "name": "ak", "definition": "a*cubed", "templateType": "anything"}, "cubed": {"description": "", "group": "Ungrouped variables", "name": "cubed", "definition": "xans^3", "templateType": "anything"}, "xans": {"description": "", "group": "Ungrouped variables", "name": "xans", "definition": "random(-12..12 except [0,1] )", "templateType": "anything"}}, "metadata": {"description": "", "notes": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "functions": {}, "name": "Simultaneous equations: simple quadratic and hyperbola, one point", "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}