// Numbas version: exam_results_page_options
{"name": "Gaussian elimination to solve a 3x3 system of linear equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Gaussian elimination to solve a 3x3 system of linear equations", "extensions": [], "variable_groups": [], "ungrouped_variables": ["a", "c", "b", "c3", "c2", "y", "x", "c1", "z"], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.</p>"}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>Re-arrange the rows so that the third row becomes the first row, the first the second and the second the third.<br/><strong>WHY?</strong> Choose one of the following:<br/>[[0]]</p>\n      <p>Now write down the entries of the matrix you will use for Gaussian Elimination, remember to include the constants as the last column.</p>\n      <table>\n      <tbody>\n      <tr>\n      <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      <td>[[1]]</td>\n      <td>[[2]]</td>\n      <td>[[3]]</td>\n      <td>[[4]]</td>\n      <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      </tr>\n      <tr>\n      <td>[[5]]</td>\n      <td>[[6]]</td>\n      <td>[[7]]</td>\n      <td>[[8]]</td>\n      </tr>\n      <tr>\n      <td>[[9]]</td>\n      <td>[[10]]</td>\n      <td>[[11]]</td>\n      <td>[[12]]</td>\n      </tr>\n      </tbody>\n      </table>\n        ", "gaps": [{"shuffleChoices": true, "matrix": [1, 0, 0, 0], "type": "1_n_2", "customMarkingAlgorithm": "", "choices": ["<p>To make sure that there is a 1 in the first row, first column position.</p>", "<p>Because you always do this.</p>", "<p>Why not.</p>", "<p>I don't know.</p>"], "scripts": {}, "displayColumns": 0, "marks": 0, "distractors": ["", "", "", ""], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "displayType": "radiogroup", "showFeedbackIcon": true, "minMarks": 0, "extendBaseMarkingAlgorithm": true, "maxMarks": 1, "showCellAnswerState": true, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "1", "scripts": {}, "marks": 0.2, 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"\n      \n      \n      <p>Now introduce zeros in the first column below the first entry by adding:<br/>[[0]] times the first row to the second row and<br/>[[1]] times the first row to the third row to get the matrix:</p>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\"> \\[\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> $\\var{b}$ </td>\n      \n      <td> $\\var{b*a-b}$ </td>\n      \n      <td> $\\var{c3}$ </td>\n      \n      <td rowspan=\"3\"> \\[\\left) \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$ </td>\n      \n      <td> [[2]] </td>\n      \n      <td> [[3]] </td>\n      \n      <td> [[4]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> [[5]] </td>\n      \n      <td> [[6]] </td>\n      \n      <td> [[7]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <p>Next multiply the second row by [[8]] to get a 1 in the second entry in the second row.</p>\n      \n      \n        ", "gaps": [{"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{-a}", "scripts": {}, "marks": 0.8, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{-a}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{-a*c}", "scripts": {}, "marks": 0.8, "notationStyles": ["plain", "en", 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false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{-a}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{c2-a*c3}", "scripts": {}, "marks": 0.6, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{c2-a*c3}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{c*b-c*b*a}", "scripts": {}, "marks": 0.6, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, 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"allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{c1-a*c*c3}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{-1}", "scripts": {}, "marks": 0.8, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{-1}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}], "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showCorrectAnswer": true, "unitTests": []}, {"prompt": "\n      \n      \n      <p>Note that you should have multiplied the second row by a suitable number to get a $1$ in the second entry in the second row.<br/>In this part we introduce a $0$ in the second column below the second entry in the second column by adding:<br/>[[0]] times the second row to the third row to get the matrix:</p>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\"> \\[\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> $\\var{b}$ </td>\n      \n      <td> $\\var{b*a-b}$ </td>\n      \n      <td> $\\var{c3}$ </td>\n      \n      <td rowspan=\"3\"> \\[\\left) \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> [[1]] </td>\n      \n      <td> [[2]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> [[3]] </td>\n      \n      <td> [[4]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <p>From this you should find:</p>\n      \n      \n      \n      <p>$z=\\;\\;$[[5]]</p>\n      \n      \n        ", "gaps": [{"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{-b*c+a*b*c}", "scripts": {}, "marks": 0.8, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{-b*c+a*b*c}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{a}", "scripts": {}, "marks": 0.8, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{a}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{a*c3-c2}", "scripts": {}, "marks": 0.8, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{a*c3-c2}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{1}", "scripts": {}, "marks": 0.8, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{1}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "scripts": {}, "marks": 0.8, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{z}", "scripts": {}, "marks": 2, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{z}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}], "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showCorrectAnswer": true, "unitTests": []}, {"prompt": "\n      \n      \n      <p>From the second row of the reduced matrix you find an equation involving only $y$ and $z$ and using your value for $z$ we find:</p>\n      \n      \n      \n      <p>$y=\\;\\;$[[0]]</p>\n      \n      \n      \n      <p>Then using the first row we have the equation :<br/>\\[\\simplify[all]{x+ {b}y+{b*a-b}z={c3}}\\]</p>\n      \n      \n      \n      <p>Using this you can now find $x$:</p>\n      \n      \n      \n      <p>$x=\\;\\;$[[1]]</p>\n      \n      \n        ", "gaps": [{"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{y}", "scripts": {}, "marks": 3, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{y}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "{x}", "scripts": {}, "marks": 2.6, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "maxValue": "{x}", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "showCorrectAnswer": true, "unitTests": []}], "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showCorrectAnswer": true, "unitTests": []}], "preamble": {"js": "", "css": ""}, "advice": "<p>Look at the revealed answers for this question. All the information needed is there.</p>", "functions": {}, "tags": [], "variables": {"c3": {"name": "c3", "templateType": "anything", "definition": "random(1..3)", "group": "Ungrouped variables", "description": ""}, "y": {"name": "y", "templateType": "anything", "definition": "a*c3-c2-a*z", "group": "Ungrouped variables", "description": ""}, "c2": {"name": "c2", "templateType": "anything", "definition": "random(1..3)", "group": "Ungrouped variables", "description": ""}, "c1": {"name": "c1", "templateType": "anything", "definition": "random(1..5)", "group": "Ungrouped variables", "description": ""}, "a": {"name": "a", "templateType": "anything", "definition": "random(2..6)", "group": "Ungrouped variables", "description": ""}, "c": {"name": "c", "templateType": "anything", "definition": "random(1..3)", "group": "Ungrouped variables", "description": ""}, "b": {"name": "b", "templateType": "anything", "definition": "random(2..6)", "group": "Ungrouped variables", "description": ""}, "z": {"name": "z", "templateType": "anything", "definition": "c1+c2*c*(b-a*b)+c3*c*(a^2*b-a-a*b)", "group": "Ungrouped variables", "description": ""}, "x": {"name": "x", "templateType": "anything", "definition": "c3-(b*a-b)*z-b*y", "group": "Ungrouped variables", "description": ""}}, "statement": "<p>Solve the system of equations using Gauss Elimination <br/>\\[\\begin{eqnarray*} &amp;\\var{a}x&amp;+\\;&amp;\\var{a*b-1}y&amp;+\\;\\var{a^2*b-a-a*b}z&amp;=&amp;\\var{c2}\\\\ &amp;\\var{a*c}x&amp;+\\;&amp;\\var{c*b}y&amp;+\\;z&amp;=&amp;\\var{c1}\\\\ &amp;x&amp;+\\;&amp;\\var{b}y&amp;+\\;\\var{b*a-b}z&amp;=&amp;\\var{c3} \\end{eqnarray*} \\] <br/><strong>Part a)</strong> Rearrange the order of the equations and represent this as a system of equations using a matrix.<br/><strong>Part b)</strong> Introduce zeros in the first column using the first row.<br/><strong>Part c)</strong> Introduce zeros in the second <g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"10\" data-gr-id=\"10\"><g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"10\" data-gr-id=\"10\"><g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"10\" data-gr-id=\"10\"><g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"10\" data-gr-id=\"10\">coumn</g></g></g></g> below the second entry in the second row using the second row. <br/><g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\"><g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\"><g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\"><g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\">Also</g></g></g></g> need to solve for $z$ using the last row of the reduced matrix.<br/><strong>Part d)</strong> Solve for $y$ and $x$ using the second and first rows of the reduced matrix.</p>", "type": "question", "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Praneetha Singh", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2552/"}]}]}], "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Praneetha Singh", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2552/"}]}