// Numbas version: exam_results_page_options
{"name": "Solve a system of three simultaneous linear equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "metadata": {"description": "<p>Solve a system of three simultaneous linear equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "extensions": [], "name": "Solve a system of three simultaneous linear equations", "statement": "<p>Solve the following system of&nbsp;three simultaneous linear equations:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x+2y+4z=\\var{r1}\\)</p>\n<p style=\"padding-left: 60px;\">and</p>\n<p style=\"padding-left: 30px;\">\\(2x+\\var{b1}y+3z=\\var{r2}\\)</p>\n<p style=\"padding-left: 60px;\">and</p>\n<p style=\"padding-left: 30px;\">\\(5x+6y+\\var{c1}z=\\var{r3}\\)</p>", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a1", "b1", "c1", "r1", "r2", "r3"], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "advice": "<p style=\"padding-left: 30px; text-align: left;\">(i) &nbsp; &nbsp;\\(\\var{a1}x+2y+4z=\\var{r1}\\)</p>\n<p style=\"padding-left: 30px;\">(ii) &nbsp; \\(2x+\\var{b1}y+3z=\\var{r2}\\)</p>\n<p style=\"padding-left: 30px;\">(iii) &nbsp;\\(5x+6y+\\var{c1}z=\\var{r3}\\)</p>\n<p>First reduce the three equations in three unknowns to a two equations in two unknowns problem by eliminating one of the variables.</p>\n<p>We can eliminate \\(x\\) using equations (i) and (ii)</p>\n<p style=\"padding-left: 30px;\">2*(i) &nbsp; &nbsp; \\(\\simplify{2*{a1}}x+4y+8z=\\simplify{2*{r1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}\\)*(ii) &nbsp; &nbsp;\\(\\simplify{2*{a1}}x+\\simplify{{a1}*{b1}}y+\\simplify{3*{a1}}z=\\simplify{{a1}*{r2}}\\)</p>\n<p>Subtracting gives us a new equation</p>\n<p style=\"padding-left: 30px;\">(iv) &nbsp; &nbsp;\\(\\simplify{(4-{a1}{b1})y+(8-3*{a1})z}=\\simplify{2*{r1}-{a1}*{r2}}\\)</p>\n<p>We can also eliminate \\(x\\) using equations (ii) and (iii)</p>\n<p style=\"padding-left: 30px;\">5*(ii) &nbsp; &nbsp;\\(10x +\\simplify{5*{b1}}y+15z=\\simplify{5*{r2}}\\)</p>\n<p style=\"padding-left: 30px;\">2*(iii) &nbsp; \\(10x+12y+\\simplify{2*{c1}}z=\\simplify{2*{r3}}\\)</p>\n<p>Subtracting gives us another new equation</p>\n<p style=\"padding-left: 30px;\">(v) &nbsp; &nbsp; \\(\\simplify{(5*{b1}-12)y+(15-2*{c1})z}=\\simplify{5*{r2}-2*{r3}}\\)</p>\n<p>We could then eliminate the \\(y\\) from these two new equations</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{5*{b1}-12}\\)*(iv) &nbsp; &nbsp;\\(\\simplify{(5*{b1}-12)*(4-{a1}{b1})y+(5*{b1}-12)*(8-3*{a1})z}=\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{4-{a1}{b1}}\\)*(v) &nbsp; &nbsp;\\(\\simplify{(4-{a1}{b1})*(5*{b1}-12)y+(4-{a1}{b1})*(15-2*{c1})z}=\\simplify{(4-{a1}{b1})*(5*{r2}-2*{r3})}\\)</p>\n<p>Subtracting gives us</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{(5*{b1}-12)*(8-3*{a1})-(4-{a1}{b1})*(15-2*{c1})}z=\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}{b1})*(5*{r2}-2*{r3})}\\)</p>\n<p>Thus</p>\n<p style=\"padding-left: 30px;\">\\(z=\\frac{\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}{b1})*(5*{r2}-2*{r3})}}{\\simplify{(5*{b1}-12)*(8-3*{a1})-(4-{a1}{b1})*(15-2*{c1})}}=\\simplify{decimal{((5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}*{b1})*(5*{r2}-2*{r3}))/( &nbsp;(5*{b1}-12)*(8-3*{a1})-(4-{a1}*{b1})*(15-2*{c1}))}}\\)</p>\n<p>We can now back substitute this value for \\(z\\) into equation (iv) to find the correct value for \\(y\\) and then&nbsp;back substitute both these values into equation (i) to calculate \\(x\\).&nbsp;</p>\n<p style=\"padding-left: 30px;\"></p>", "tags": [], "variables": {"b1": {"templateType": "randrange", "description": "", "definition": "random(2..10#1)", "group": "Ungrouped variables", "name": "b1"}, "r1": {"templateType": "randrange", "description": "", "definition": "random(20..42#1)", "group": "Ungrouped variables", "name": "r1"}, "a1": {"templateType": "randrange", "description": "", "definition": "random(2..8#1)", "group": "Ungrouped variables", "name": "a1"}, "c1": {"templateType": "randrange", "description": "", "definition": "random(3..12#1)", "group": "Ungrouped variables", "name": "c1"}, "r3": {"templateType": "randrange", "description": "", "definition": "random(30..60#1)", "group": "Ungrouped variables", "name": "r3"}, "r2": {"templateType": "randrange", "description": "", "definition": "random(18..50#1)", "group": "Ungrouped variables", "name": "r2"}}, "type": "question", "parts": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": [], "prompt": "<p>Input the value of \\(x\\) that satisfies the three equations.</p>\n<p style=\"padding-left: 30px;\">\\(x = \\)&nbsp;[[0]]</p>\n<p>Input the value of \\(y\\) that satisfies the three equations.</p>\n<p style=\"padding-left: 30px;\">\\(y = \\)&nbsp;[[1]]</p>\n<p>Input the value of \\(z\\) that satisfies the three equations.</p>\n<p style=\"padding-left: 30px;\">\\(z = \\)&nbsp;[[2]]</p>", "gaps": [{"marks": 1, "precisionPartialCredit": 0, "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "precisionType": "dp", "minValue": "((18-{b1}*{c1})*(3*{r1}-{r3})+({c1}-12)*(6*{r2}-{b1}*{r3}))/((3*{a1}-5)*(18-{b1}*{c1})-(12-5*{b1})*(12-{c1}))", "showPrecisionHint": true, "showCorrectAnswer": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "maxValue": "((18-{b1}*{c1})*(3*{r1}-{r3})+({c1}-12)*(6*{r2}-{b1}*{r3}))/((3*{a1}-5)*(18-{b1}*{c1})-(12-5*{b1})*(12-{c1}))"}, {"marks": 1, "precisionPartialCredit": 0, "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "precisionType": "dp", "minValue": "((15-2*{c1})*(2*{r1}-{a1}*{r2})+(3*{a1}-8)*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "showPrecisionHint": true, "showCorrectAnswer": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "maxValue": "((15-2*{c1})*(2*{r1}-{a1}*{r2})+(3*{a1}-8)*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))"}, {"marks": 1, "precisionPartialCredit": 0, "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "precisionType": "dp", "minValue": "((12-5*{b1})*(2*{r1}-{a1}*{r2})+(4-{b1}*{a1})*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "showPrecisionHint": true, "showCorrectAnswer": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "maxValue": "((12-5*{b1})*(2*{r1}-{a1}*{r2})+(4-{b1}*{a1})*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))"}], "showFeedbackIcon": true, "type": "gapfill", "showCorrectAnswer": true}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Thomas Waters", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3649/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Thomas Waters", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3649/"}]}