// Numbas version: finer_feedback_settings
{"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": [{"name": "Simultaneous equations: linear and quadratic, one point", "tags": ["algebra", "Algebra", "equations", "quadratic", "Simultaneous equations", "simultaneous equations", "solving equations", "Solving equations", "system of equations"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"quadxcoeff": {"name": "quadxcoeff", "group": "Ungrouped variables", "definition": "sroots+grad", "description": "", "templateType": "anything", "can_override": false}, "ansyvalue": {"name": "ansyvalue", "group": "Ungrouped variables", "definition": "grad*root1+yint", "description": "", "templateType": "anything", "can_override": false}, "yint": {"name": "yint", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "root1": {"name": "root1", "group": "Ungrouped variables", "definition": "random(-12..12)", "description": "", "templateType": "anything", "can_override": false}, "proots": {"name": "proots", "group": "Ungrouped variables", "definition": "root1*root2", "description": "", "templateType": "anything", "can_override": false}, "quadccoeff": {"name": "quadccoeff", "group": "Ungrouped variables", "definition": "proots+yint", "description": "", "templateType": "anything", "can_override": false}, "sroots": {"name": "sroots", "group": "Ungrouped variables", "definition": "-root1-root2", "description": "", "templateType": "anything", "can_override": false}, "grad": {"name": "grad", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "root2": {"name": "root2", "group": "Ungrouped variables", "definition": "root1", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["root1", "root2", "grad", "sroots", "proots", "yint", "quadxcoeff", "quadccoeff", "ansyvalue"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "root1", "maxValue": "root1", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansyvalue", "maxValue": "ansyvalue", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "resources": []}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}