// 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": [{"functions": {}, "ungrouped_variables": ["xans", "cubed", "a", "ak", "b", "yans"], "name": "Simultaneous equations: simple quadratic and hyperbola, one point", "tags": ["algebra", "equations", "hyperbola", "quadratic", "Simultaneous equations", "simultaneous equations", "solving equations", "Solving equations", "system of equations"], "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

Find the $x$ and $y$ values that satisfy both of the following equations. That is, find the point of intersection of the two curves.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$y$$=$$\\simplify{{ak}/({b}x)}$               $(1)$
$y$$=$$\\simplify{{a}x^2/{b}}$               $(2)$
\n

$x=$ [[0]],   $y=$ [[1]]

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "xans", "minValue": "xans", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": true, "variableReplacements": [], "maxValue": "a*xans^2/b", "minValue": "a*xans^2/b", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "

Given

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$y$$=$$\\simplify{{ak}/({b}x)}$               $(1)$
$y$$=$$\\simplify{{a}x^2/{b}}$               $(2)$
\n

substitute the expression for $y$ given in $(1)$ into $(2)$:
\\[\\simplify{{ak}/({b}x) ={a}x^2/{b}}\\]

\n

To get rid of the $x$ in the denominator, let us multiply both sides by $x$
\\[\\simplify{{ak}/({b}) ={a}x^3/{b}}\\]

\n

Since there is only one term with an $x$ in it, we can get $x^3$ by itself

\n

\\[x^3=\\var{cubed}\\]

\n

Therefore, $x=\\sqrt[3]{\\var{cubed}}=\\var{xans}$.

\n


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 $(2)$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
\n

$y$

\n
\n

$=$

\n
\n

$\\simplify{{a}/{b}}(\\var{xans})^2$

\n
\n

$=$

\n
\n

$\\simplify{{a*xans^2/b}}$

\n
\n

Therefore the values that satisfy equations $(1)$ and $(2)$ are $x=\\var{xans}$ and $y=\\simplify{{a*xans^2/b}}$.

\n

 

", "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(-4..4 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-12..12 except [a,0,1,-1])", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "yans": {"definition": "a*cubed/b", "templateType": "anything", "group": "Ungrouped variables", "name": "yans", "description": ""}, "xans": {"definition": "random(-12..12 except [0,1] )", "templateType": "anything", "group": "Ungrouped variables", "name": "xans", "description": ""}, "ak": {"definition": "a*cubed", "templateType": "anything", "group": "Ungrouped variables", "name": "ak", "description": ""}, "cubed": {"definition": "xans^3", "templateType": "anything", "group": "Ungrouped variables", "name": "cubed", "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/"}]}