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{"name": "Simultaneous equations: linear and quadratic, one point", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"showQuestionGroupNames": false, "name": "Simultaneous equations: linear and quadratic, one point", "metadata": {"notes": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": ""}, "statement": "", "functions": {}, "ungrouped_variables": ["root1", "root2", "grad", "sroots", "proots", "yint", "quadxcoeff", "quadccoeff", "ansyvalue"], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"root1": {"templateType": "anything", "name": "root1", "description": "", "group": "Ungrouped variables", "definition": "random(-12..12)"}, "quadccoeff": {"templateType": "anything", "name": "quadccoeff", "description": "", "group": "Ungrouped variables", "definition": "proots+yint"}, "grad": {"templateType": "anything", "name": "grad", "description": "", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)"}, "ansyvalue": {"templateType": "anything", "name": "ansyvalue", "description": "", "group": "Ungrouped variables", "definition": "grad*root1+yint"}, "yint": {"templateType": "anything", "name": "yint", "description": "", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)"}, "proots": {"templateType": "anything", "name": "proots", "description": "", "group": "Ungrouped variables", "definition": "root1*root2"}, "root2": {"templateType": "anything", "name": "root2", "description": "", "group": "Ungrouped variables", "definition": "root1"}, "quadxcoeff": {"templateType": "anything", "name": "quadxcoeff", "description": "", "group": "Ungrouped variables", "definition": "sroots+grad"}, "sroots": {"templateType": "anything", "name": "sroots", "description": "", "group": "Ungrouped variables", "definition": "-root1-root2"}}, "preamble": {"js": "", "css": ""}, "tags": ["algebra", "equations", "quadratic", "Simultaneous equations", "simultaneous equations", "solving equations", "Solving equations", "system of equations"], "variable_groups": [], "type": "question", "advice": "", "question_groups": [{"pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered", "questions": []}], "parts": [{"variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "correctAnswerFraction": false, "showPrecisionHint": false, "maxValue": "root1", "variableReplacements": [], "type": "numberentry", "allowFractions": true, "showCorrectAnswer": true, "minValue": "root1", "marks": 1}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "correctAnswerFraction": true, "showPrecisionHint": false, "maxValue": "ansyvalue", "variableReplacements": [], "type": "numberentry", "allowFractions": true, "showCorrectAnswer": true, "minValue": "ansyvalue", "marks": 1}], "scripts": {}, "variableReplacements": [], "stepsPenalty": 0, "marks": 0, "type": "gapfill", "showCorrectAnswer": true, "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;\">$\\simplify{{grad}x+{yint}}$</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>$x=$&nbsp;[[0]], &nbsp; $y=$&nbsp;[[1]]</p>\n<p></p>", "steps": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true, "type": "information", "marks": 0, "variableReplacements": [], "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;$x=\\var{root1}$.</p>\n<p><br/>Now we know the $x$ value 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;\">$\\var{ansyvalue}$</td>\n</tr>\n</tbody>\n</table>\n<p>Therefore the values that satisfy equations $(1)$ and $(2)$ are $x=\\var{root1}$ and $y=\\var{ansyvalue}$.</p>"}]}], "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/"}]}