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{"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": [{"parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "marks": 0, "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-{minroot}})(\\simplify{x-{maxroot}})=0\\]</p>\n<p></p>\n<p>Therefore,&nbsp; $x_1=\\var{minroot}$,&nbsp; $x_2=\\var{maxroot}$.</p>\n<p><br/>Now we know the $x$ values we can determine the corresponding $y$ values by substituting $x_1=\\var{minroot}$&nbsp; and $x_2=\\var{maxroot}$&nbsp; into either equation $(1)$ or $(2)$, below we substitute into $(1)$:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y_1$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!collectnumbers]{{grad}({minroot})+{yint}}$</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<tr>\n<td style=\"text-align: right;\">$y_2$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!collectnumbers]{{grad}({maxroot})+{yint}}$</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;\">$\\var{ansyvalue2}$</td>\n</tr>\n</tbody>\n</table>\n<p>Therefore the values that satisfy equations $(1)$ and $(2)$ are $x=\\var{minroot}$ and $y=\\var{ansyvalue}$ or $x=\\var{maxroot}$ and $y=\\var{ansyvalue2}$</p>", "variableReplacements": [], "scripts": {}}], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "stepsPenalty": 0, "gaps": [{"mustBeReducedPC": 0, "allowFractions": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "maxValue": "minroot", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "minValue": "minroot", "showCorrectAnswer": true, "mustBeReduced": false, "scripts": {}}, {"mustBeReducedPC": 0, "allowFractions": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "maxValue": "ansyvalue", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true, "correctAnswerFraction": true, "type": "numberentry", "minValue": "ansyvalue", "showCorrectAnswer": true, "mustBeReduced": false, "scripts": {}}, {"mustBeReducedPC": 0, "allowFractions": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "maxValue": "maxroot", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "minValue": "maxroot", "showCorrectAnswer": true, "mustBeReduced": false, "scripts": {}}, {"mustBeReducedPC": 0, "allowFractions": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "maxValue": "ansyvalue2", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true, "correctAnswerFraction": true, "type": "numberentry", "minValue": "ansyvalue2", "showCorrectAnswer": true, "mustBeReduced": false, "scripts": {}}], "marks": 0, "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>$x=$&nbsp;[[2]], &nbsp; $y=$&nbsp;[[3]]</p>\n<p></p>", "variableReplacements": [], "scripts": {}}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"minroot": {"name": "minroot", "group": "Ungrouped variables", "definition": "min(root1,root2)", "templateType": "anything", "description": ""}, "quadxcoeff": {"name": "quadxcoeff", "group": "Ungrouped variables", "definition": "sroots+grad", "templateType": "anything", "description": ""}, "ansyvalue2": {"name": "ansyvalue2", "group": "Ungrouped variables", "definition": "grad*maxroot+yint", "templateType": "anything", "description": ""}, "sroots": {"name": "sroots", "group": "Ungrouped variables", "definition": "-root1-root2", "templateType": "anything", "description": ""}, "root2": {"name": "root2", "group": "Ungrouped variables", "definition": "random(-12..12)", "templateType": "anything", "description": ""}, "root1": {"name": "root1", "group": "Ungrouped variables", "definition": "random(-12..12)", "templateType": "anything", "description": ""}, "quadccoeff": {"name": "quadccoeff", "group": "Ungrouped variables", "definition": "proots+yint", "templateType": "anything", "description": ""}, "maxroot": {"name": "maxroot", "group": "Ungrouped variables", "definition": "max(root1, root2)", "templateType": "anything", "description": ""}, "proots": {"name": "proots", "group": "Ungrouped variables", "definition": "root1*root2", "templateType": "anything", "description": ""}, "yint": {"name": "yint", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "templateType": "anything", "description": ""}, "grad": {"name": "grad", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "templateType": "anything", "description": ""}, "ansyvalue": {"name": "ansyvalue", "group": "Ungrouped variables", "definition": "grad*minroot+yint", "templateType": "anything", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": ""}, "extensions": [], "functions": {}, "name": "Simultaneous equations: linear and quadratic, two points", "variable_groups": [], "tags": [], "rulesets": {}, "ungrouped_variables": ["root1", "root2", "grad", "sroots", "proots", "yint", "quadxcoeff", "quadccoeff", "ansyvalue", "ansyvalue2", "minroot", "maxroot"], "statement": "", "advice": "", "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Katherine Tomlinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1652/"}]}]}], "contributors": [{"name": "Katherine Tomlinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1652/"}]}