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{"name": "Gauss-Seidel method #2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Gauss-Seidel method #2", "statement": "<p>Perform the first three iterations of the Jacobi method on the system of simultaneous linear equations below, taking&nbsp; &nbsp;\\(y_0=\\var{y0},\\,\\,\\,z_0=\\var{z0}\\) as your initial guess.</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x+\\var{a2}y+\\var{a3}z=\\var{r}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a4}x+\\var{a5}y+\\var{a6}z=\\var{s}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a7}x+\\var{a8}y+\\var{a9}z=\\var{t}\\)</p>\n<p>Give all your answers correct to three decimal places.</p>", "preamble": {"css": "", "js": ""}, "rulesets": {}, "extensions": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "tags": [], "advice": "<p>Rearranging the equations gives the following:</p>\n<p style=\"padding-left: 30px;\">\\(x_{n+1}=\\frac{1}{\\var{a1}}(\\var{r}-\\var{a2}y_n-\\var{a3}z_n)\\)</p>\n<p style=\"padding-left: 30px;\">\\(y_{n+1}=\\frac{1}{\\var{a5}}(\\var{s}-\\var{a4}x_{n+1}-\\var{a6}z_n)\\)</p>\n<p style=\"padding-left: 30px;\">\\(z_{n+1}=\\frac{1}{\\var{a9}}(\\var{t}-\\var{a7}x_{n+1}-\\var{a8}y_{n+1})\\)</p>\n<p>\\(y_0=\\var{y0},\\,\\,\\,z_0=\\var{z0}\\)</p>\n<p></p>\n<p style=\"padding-left: 30px;\">\\(x_1=\\frac{1}{\\var{a1}}(\\var{r}-\\var{a2}(\\var{y0})-\\var{a3}(\\var{z0}))=\\frac{1}{\\var{a1}}(\\simplify{{r}-{a2}*{y0}-{a3}*{z0}})=\\var{x1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(y_1=\\frac{1}{\\var{a5}}(\\var{s}-\\var{a4}(\\var{x1})-\\var{a6}(\\var{z0}))=\\frac{1}{\\var{a5}}(\\simplify{{s}-{a4}*{x1}-{a6}*{z0}})=\\var{y1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(z_1=\\frac{1}{\\var{a9}}(\\var{t}-\\var{a7}(\\var{x1})-\\var{a8}(\\var{y1}))=\\frac{1}{\\var{a9}}(\\simplify{{t}-{a7}*{x1}-{a8}*{y1}})=\\var{z1}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>\\(x_1=\\var{x1},\\,\\,\\,y_1=\\var{y1},\\,\\,\\,z_1=\\var{z1}\\)</p>\n<p></p>\n<p style=\"padding-left: 30px;\">\\(x_2=\\frac{1}{\\var{a1}}(\\var{r}-\\var{a2}(\\var{y1})-\\var{a3}(\\var{z1}))=\\frac{1}{\\var{a1}}(\\simplify{{r}-{a2}*{y1}-{a3}*{z1}})=\\var{x2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(y_2=\\frac{1}{\\var{a5}}(\\var{s}-\\var{a4}(\\var{x2})-\\var{a6}(\\var{z1}))=\\frac{1}{\\var{a5}}(\\simplify{{s}-{a4}*{x2}-{a6}*{z1}})=\\var{y2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(z_2=\\frac{1}{\\var{a9}}(\\var{t}-\\var{a7}(\\var{x2})-\\var{a8}(\\var{y2}))=\\frac{1}{\\var{a9}}(\\simplify{{t}-{a7}*{x2}-{a8}*{y2}})=\\var{z2}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>\\(x_2=\\var{x2},\\,\\,\\,y_2=\\var{y2},\\,\\,\\,z_2=\\var{z2}\\)</p>\n<p></p>\n<p style=\"padding-left: 30px;\">\\(x_3=\\frac{1}{\\var{a1}}(\\var{r}-\\var{a2}(\\var{y2})-\\var{a3}(\\var{z2}))=\\frac{1}{\\var{a1}}(\\simplify{{r}-{a2}*{y2}-{a3}*{z2}})=\\var{x3}\\)</p>\n<p style=\"padding-left: 30px;\">\\(y_3=\\frac{1}{\\var{a5}}(\\var{s}-\\var{a4}(\\var{x3})-\\var{a6}(\\var{z2}))=\\frac{1}{\\var{a5}}(\\simplify{{s}-{a4}*{x3}-{a6}*{z2}})=\\var{y3}\\)</p>\n<p style=\"padding-left: 30px;\">\\(z_3=\\frac{1}{\\var{a9}}(\\var{t}-\\var{a7}(\\var{x3})-\\var{a8}(\\var{y3}))=\\frac{1}{\\var{a9}}(\\simplify{{t}-{a7}*{x3}-{a8}*{y3}})=\\var{z3}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "variable_groups": [{"name": "Unnamed group", "variables": []}], "variables": {"a5": {"name": "a5", "definition": "random(9..15#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "a7": {"name": "a7", "definition": "random(2..6#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "x3": {"name": "x3", "definition": "({r}-{a2}*{y2}-{a3}*{z2})/{a1}", 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