// Numbas version: finer_feedback_settings
{"name": "AC5 Simultaneous Equations (one non-linear)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "AC5 Simultaneous Equations (one non-linear)", "tags": [], "metadata": {"description": "<p>Solving a pair of simultaneous equations of the form $a_1x+y=c_1$ and $a_2x^2+b_2xy=c_2$ by forming a quadratic equation.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the following simultaneous equations:</p>\n<p>\\[ \\begin{split} \\simplify{{a1}x+y} &amp;\\,= \\var{c1} \\\\ \\simplify{{a2}x^2+{b2}x*y} &amp;\\,= \\var{c2} \\end{split} \\]</p>\n<p></p>\n<p><em>Give your answers to 2 decimal places where necessary.</em></p>", "advice": "<p>To solve a pair of simultaneous equations of this type we want to rearrange the linear equation such that $y$ is the subject, which we can then substitute into the equation with the quadratic $x$-term. This will result in a quadratic equation in terms of $x$ only.</p>\n<p>For the equations&nbsp;</p>\n<p>\\[ \\begin{split} \\simplify{{a1}x+y} &amp;\\,= \\var{c1} \\qquad \\qquad &amp;(1) \\\\\\simplify{{a2}x^2+{b2}x*y} &amp;\\,= \\var{c2} \\qquad \\qquad &amp;(2) \\end{split} \\]</p>\n<p>we can rearrange equation (1)&nbsp;to make $y$ the subject:</p>\n<p>\\[ y = \\simplify{{c1}-{a1}x}. \\qquad\\qquad (3)\\]</p>\n<p>Substituting this into equation (2):</p>\n<p>\\[ \\begin{split}\\simplify{{a2}x^2+{b2}x({c1}-{a1}x)} &amp;\\,=\\var{c2} \\\\ \\simplify[!cancelTerms,unitFactor]{{a2}x^2+{b2*c1}x-{b2*a1}x^2} &amp;\\,=\\var{c2}. \\end{split} \\]</p>\n<p>Collecting similar terms:</p>\n<p>\\[ \\simplify{({a2}-{b2*a1})x^2+{b2*c1}x-{c2}} =0. \\qquad\\qquad (4) \\]</p>\n<p>Using the quadratic formula, we find two solutions for $x$:</p>\n<p>{check}</p>\n<p>Therefore, the 2 pairs of solutions for these simultaneous equations are</p>\n<p>\\[ (x_1,y_1) = (\\var{x1dp},\\var{y1dp}) \\] and&nbsp;\\[ (x_2,y_2) = (\\var{x2dp},\\var{y2dp}). \\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Rearranging\"><span style=\"font-weight: 400;\"></span></a><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Sim\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-5..-1)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "solx1": {"name": "solx1", "group": "Ungrouped variables", "definition": "(-b2*c1-sqrt((b2^2*c1^2)+4(a2-a1*b2)*c2))/(2(a2-b2*a1))", "description": "", "templateType": "anything", "can_override": false}, "solx2": {"name": "solx2", "group": "Ungrouped variables", "definition": "(-b2*c1+sqrt((b2^2*c1^2)+4(a2-a1*b2)*c2))/(2(a2-b2*a1))", "description": "", "templateType": "anything", "can_override": false}, "soly1": {"name": "soly1", "group": "Ungrouped variables", "definition": "c1-a1*solx1", "description": "", "templateType": "anything", "can_override": false}, "soly2": {"name": "soly2", "group": "Ungrouped variables", "definition": "c1-a1*solx2", "description": "", "templateType": "anything", "can_override": false}, "x2dp": {"name": "x2dp", "group": "Ungrouped variables", "definition": "precround(solx2,2)", "description": "", "templateType": "anything", "can_override": false}, "y1dp": {"name": "y1dp", "group": "Ungrouped variables", "definition": "precround(soly1,2)", "description": "", "templateType": "anything", "can_override": false}, "y2dp": {"name": "y2dp", "group": "Ungrouped variables", "definition": "precround(soly2,2)", "description": "", "templateType": "anything", "can_override": false}, "x1dp": {"name": "x1dp", "group": "Ungrouped variables", "definition": "precround(solx1,2)", "description": "", "templateType": "anything", "can_override": false}, "solutions1": {"name": "solutions1", "group": "Ungrouped variables", "definition": "matrix([x1dp,y1dp])", "description": "", "templateType": "anything", "can_override": false}, "solutions2": {"name": "solutions2", "group": "Ungrouped variables", "definition": "matrix([x2dp,y2dp])", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Ungrouped variables", "definition": "if(x1dp=round(x1dp) and x2dp=round(x2dp),'{text}', if(x1dp=round(x1dp),'{text1}',if(x2dp=round(x2dp),'{text2}','{text3}')))", "description": "", "templateType": "anything", "can_override": false}, "text1": {"name": "text1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ x_1 = \\\\var{x1dp} \\\\,&nbsp; \\\\quad \\\\text{and} \\\\quad x_2=\\\\var{x2dp} \\\\, \\\\text{ (2 d.p.)} \\\\]</p>\\n<p>To find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:</p>\\n<p>\\\\[ y_1 = \\\\var{y1dp} \\\\,&nbsp; \\\\quad \\\\text{and} \\\\quad y_2=\\\\var{y2dp} \\\\, \\\\text{(2 d.p.)} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "text2": {"name": "text2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ x_1 = \\\\var{x1dp} \\\\, \\\\text{ (2 d.p.)}\\\\quad \\\\text{and} \\\\quad x_2=\\\\var{x2dp} \\\\]</p>\\n<p>To find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:</p>\\n<p>\\\\[ y_1 = \\\\var{y1dp} \\\\, \\\\text{(2 d.p.)} \\\\quad \\\\text{and} \\\\quad y_2=\\\\var{y2dp} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "text3": {"name": "text3", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ x_1 = \\\\var{x1dp} \\\\,&nbsp; \\\\text{ (2 d.p.)}\\\\quad \\\\text{and} \\\\quad x_2=\\\\var{x2dp} \\\\, \\\\text{(2 d.p.)} \\\\]</p>\\n<p>To find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:</p>\\n<p>\\\\[ y_1 = \\\\var{y1dp} \\\\, \\\\text{(2 d.p.)} \\\\quad \\\\text{and} \\\\quad y_2=\\\\var{y2dp} \\\\,&nbsp;\\\\text{(2 d.p.)} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "text": {"name": "text", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ x_1 = \\\\var{x1dp} \\\\,&nbsp; \\\\quad \\\\text{and} \\\\quad x_2=\\\\var{x2dp}\\\\]</p>\\n<p>To find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:</p>\\n<p>\\\\[ y_1 = \\\\var{y1dp} \\\\,&nbsp; \\\\quad \\\\text{and} \\\\quad y_2=\\\\var{y2dp} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}}, "variablesTest": {"condition": "(a2-a1*b2)>0 and (b2^2*c1^2+4(a2-a1*b2)*c2)>0 and gcd(a2,b2)=1", "maxRuns": 100}, "ungrouped_variables": ["a1", "c1", "a2", "b2", "c2", "solx1", "solx2", "soly1", "soly2", "x1dp", "y1dp", "x2dp", "y2dp", "solutions1", "solutions2", "check", "text", "text1", "text2", "text3"], "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>$(x_1,y_1)=$[[0]]</p>\n<p>$(x_2,y_2)=$[[1]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "correctAnswer": "solutions2", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "correctAnswer": "solutions1", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "correctAnswer": "solutions1", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "correctAnswer": "solutions2", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}]}