// Numbas version: exam_results_page_options
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"metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "name": "Gauss-Seidel method", "parts": [{"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "x1", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "55", "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "x1", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "y1", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "y1", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "z1", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "z1", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "x2", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "x2", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "y2", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "y2", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "z2", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "z2", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "x3", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "x3", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "y3", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "y3", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "z3", "showCorrectAnswer": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showPrecisionHint": false, "correctAnswerFraction": false, "precision": "3", "strictPrecision": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "z3", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}], "prompt": "<p>\\(y_0=\\var{y0}\\) &nbsp; and &nbsp; \\(z_0=\\var{z0}\\)</p>\n<p>The first iteration gives:</p>\n<p style=\"padding-left: 30px;\">\\(x_1=\\)&nbsp;[[0]]</p>\n<p style=\"padding-left: 30px;\">\\(y_1=\\)&nbsp;[[1]]</p>\n<p style=\"padding-left: 30px;\">\\(z_1=\\)&nbsp;[[2]]</p>\n<p>The second iteration gives:</p>\n<p style=\"padding-left: 30px;\">\\(x_2=\\)&nbsp;[[3]]</p>\n<p style=\"padding-left: 30px;\">\\(y_2=\\)&nbsp;[[4]]</p>\n<p style=\"padding-left: 30px;\">\\(z_2=\\)&nbsp;[[5]]</p>\n<p>The third iteration gives:</p>\n<p style=\"padding-left: 30px;\">\\(x_3=\\)&nbsp;[[6]]</p>\n<p style=\"padding-left: 30px;\">\\(y_3=\\)&nbsp;[[7]]</p>\n<p style=\"padding-left: 30px;\">\\(z_3=\\)&nbsp;[[8]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "statement": "<p>Perform the first three iterations in the Gauss-Seidel method on the following system of equations:</p>\n<p style=\"padding-left: 60px;\">\\(\\var{a1}x+\\var{b1}y+\\var{c1}z=\\var{r1}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\var{a2}x+\\var{b2}y+\\var{c2}z=\\var{r2}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\var{a3}x+\\var{b3}y+\\var{c3}z=\\var{r3}\\)</p>\n<p>taking &nbsp; \\(y_0=\\var{y0}\\) &nbsp; and &nbsp; \\(z_0=\\var{z0}\\)</p>\n<p>Give all your answers correct to three decimal places.</p>", "variable_groups": [], "rulesets": {}, "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["y3", "a1", "b1", "c1", "a2", "b2", "c2", "a3", "b3", "c3", "x", "y", "z", "r1", "r2", "est", "y0", "r3", "z0", "x1", "y1", "z1", "x2", "y2", "z2", "x3", "z3"], "advice": "<p style=\"padding-left: 60px;\">\\(\\var{a1}x+\\var{b1}y+\\var{c1}z=\\var{r1}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\var{a2}x+\\var{b2}y+\\var{c2}z=\\var{r2}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\var{a3}x+\\var{b3}y+\\var{c3}z=\\var{r3}\\)</p>\n<p>Rearranging the equations gives:</p>\n<p style=\"padding-left: 60px;\">\\(x_{n+1}=\\frac{1}{\\var{a1}}\\left(\\var{r1}-\\var{b1}y_n-\\var{c1}z_n\\right)\\)</p>\n<p style=\"padding-left: 60px;\">\\(y_{n+1}=\\frac{1}{\\var{b2}}\\left(\\var{r2}-\\var{a2}x_{n+1}-\\var{c2}z_n\\right)\\)</p>\n<p style=\"padding-left: 60px;\">\\(z_{n+1}=\\frac{1}{\\var{c3}}\\left(\\var{r3}-\\var{a3}x_{n+1}-\\var{b3}y_{n+1}\\right)\\)</p>\n<p>\\(y_0=\\var{y0}\\) &nbsp; and &nbsp; \\(z_0=\\var{z0}\\)</p>\n<p style=\"padding-left: 60px;\">\\(x_{1}=\\frac{1}{\\var{a1}}\\left(\\var{r1}-\\var{b1}*(\\var{y0})-\\var{c1}(\\var{z0})\\right)=\\var{x1}\\)</p>\n<p style=\"padding-left: 60px;\">\\(y_{1}=\\frac{1}{\\var{b2}}\\left(\\var{r2}-\\var{a2}*(\\var{x1})-\\var{c2}(\\var{z0})\\right)=\\var{y1}\\)</p>\n<p style=\"padding-left: 60px;\">\\(z_{1}=\\frac{1}{\\var{c3}}\\left(\\var{r3}-\\var{a3}*(\\var{x1})-\\var{b3}(\\var{y1})\\right)=\\var{z1}\\)</p>\n<p style=\"padding-left: 60px;\"></p>\n<p style=\"padding-left: 60px;\">\\(x_{2}=\\frac{1}{\\var{a1}}\\left(\\var{r1}-\\var{b1}*(\\var{y1})-\\var{c1}(\\var{z1})\\right)=\\var{x2}\\)</p>\n<p style=\"padding-left: 60px;\">\\(y_{2}=\\frac{1}{\\var{b2}}\\left(\\var{r2}-\\var{a2}*(\\var{x2})-\\var{c2}(\\var{z1})\\right)=\\var{y2}\\)</p>\n<p style=\"padding-left: 60px;\">\\(z_{2}=\\frac{1}{\\var{c3}}\\left(\\var{r3}-\\var{a3}*(\\var{x2})-\\var{b3}(\\var{y2})\\right)=\\var{z2}\\)</p>\n<p style=\"padding-left: 60px;\"></p>\n<p style=\"padding-left: 60px;\">\\(x_{3}=\\frac{1}{\\var{a1}}\\left(\\var{r1}-\\var{b1}*(\\var{y2})-\\var{c1}(\\var{z2})\\right)=\\var{x3}\\)</p>\n<p style=\"padding-left: 60px;\">\\(y_{3}=\\frac{1}{\\var{b2}}\\left(\\var{r2}-\\var{a2}*(\\var{x3})-\\var{c2}(\\var{z2})\\right)=\\var{y3}\\)</p>\n<p style=\"padding-left: 60px;\">\\(z_{3}=\\frac{1}{\\var{c3}}\\left(\\var{r3}-\\var{a3}*(\\var{x3})-\\var{b3}(\\var{y3})\\right)=\\var{z3}\\)</p>", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}