// Numbas version: exam_results_page_options
{"name": "Polynomials: Equating coefficients 2 (simultaneous equations) [L4 Randomised]", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"ycans": {"group": "part c", "name": "ycans", "templateType": "anything", "definition": "(q*p-s*m)/(r*m-n*q)", "description": ""}, "r": {"group": "part c", "name": "r", "templateType": "anything", "definition": "random(-12..12 except [0,ceil(n*q/m),floor(n*q/m)])", "description": ""}, "xcans": {"group": "part c", "name": "xcans", "templateType": "anything", "definition": "(n*s-p*r)/(r*m-n*q)", "description": ""}, "p": {"group": "part c", "name": "p", "templateType": "anything", "definition": "random(-12..12 except 0)", "description": ""}, "s": {"group": "part c", "name": "s", "templateType": "anything", "definition": "random(-12..12 except 0)", "description": ""}, "n": {"group": "part c", "name": "n", "templateType": "anything", "definition": "random(-12..12 except 0)", "description": ""}, "m": {"group": "part c", "name": "m", "templateType": "anything", "definition": "random(-12..12 except 0)", "description": ""}, "q": {"group": "part c", "name": "q", "templateType": "anything", "definition": "random(-12..12 except 0)", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Very good feedback and corresponds to instance of randomisation</p>"}, "tags": ["l4", "L4"], "extensions": [], "preamble": {"js": "", "css": ""}, "variable_groups": [{"variables": ["m", "n", "p", "q", "r", "s", "xcans", "ycans"], "name": "part c"}], "statement": "<p>Suppose that during some working, you find that</p>\n<p>\\[\\simplify{{-s}x+{-p}={q}*A*x+{r}*B*x+{m}A+{n}B}\\]</p>\n<p>for all $x$.</p>", "ungrouped_variables": [], "parts": [{"showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "type": "gapfill", "scripts": {}, "showFeedbackIcon": true, "prompt": "<p>By equating coefficients, determine the values of $A$ and $B$.</p>\n<p>$A=$&nbsp;[[0]], &nbsp; $B=$&nbsp;[[1]]</p>\n<p></p>", "variableReplacements": [], "sortAnswers": false, "marks": 0, "unitTests": [], "gaps": [{"minValue": "xcans", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "type": "numberentry", "scripts": {}, "showFeedbackIcon": true, "unitTests": [], "mustBeReducedPC": 0, "maxValue": "xcans", "correctAnswerFraction": true, "variableReplacements": [], "allowFractions": true, "marks": 1, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"minValue": "ycans", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "type": "numberentry", "scripts": {}, "showFeedbackIcon": true, "unitTests": [], "mustBeReducedPC": 0, "maxValue": "ycans", "correctAnswerFraction": true, "variableReplacements": [], "allowFractions": true, "marks": 1, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "variableReplacementStrategy": "originalfirst"}], "advice": "<p>Two polynomials are equal if and only if their corresponding coefficients&nbsp;are also&nbsp;equal.</p>\n<p>So we 'equate the coefficients' of the polynomials given in this question, this means that the constant terms must be equal, ie</p>\n<p></p>\n<p>\\[\\simplify{{-p}={m}*A+{n}*B}\\qquad\\qquad(1)\\]</p>\n<p>the coefficients of the $x$ terms must be equal, ie</p>\n<p>\\[\\simplify{{-s}={q}A+{r}*B}\\qquad\\qquad(2)\\]</p>\n<p></p>\n<p>We now solve these equations simultaneously and find&nbsp;$A=\\simplify[fractionnumbers]{{xcans}}$ and $B=\\simplify[fractionnumbers]{{ycans}}$.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>The working for simulations equations <g class=\"gr_ gr_14 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"14\" data-gr-id=\"14\">is</g> given below.</p>\n<p>There are many ways to solve these equations simultaneously. Here is one method.</p>\n<table>\n<tbody>\n<tr>\n<td>$\\simplify{{m}A+{n}B}$</td>\n<td>$=$</td>\n<td>$\\var{-p}$</td>\n<td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(1)$</td>\n</tr>\n<tr>\n<td>$\\simplify{{q}A+{r}B}$</td>\n<td>$=$</td>\n<td>$\\var{-s}$</td>\n<td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(2)$</td>\n</tr>\n</tbody>\n</table>\n<p>Solve one of the equations for one of the variables. Here we solve equation $(1)$ for $B$:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\var{n}B$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{-m}A+{-p}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>&nbsp;</td>\n<td></td>\n<td></td>\n</tr>\n<tr>\n<td>$B$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify{({-m}A+{-p})/({n})}}$</td>\n<td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(3)$</td>\n</tr>\n</tbody>\n</table>\n<p>Substitute this expression for $B$ given in $(3)$ into $(2)$:</p>\n<p>\\[\\simplify[all,!collectnumbers]{{q}A+{r}*(({-m}A+{-p})/{n}) =-&nbsp;{s}}\\]</p>\n<p>Collect like terms:<br/>\\[\\simplify[fractionnumbers]{{q-r*m/n}A={-s+r*p/n}}\\]</p>\n<p>Solve for $A$:<br/>\\[A=\\simplify[fractionnumbers]{{xcans}}\\]<br/><br/>Now we know the $A$ value we can determine the corresponding $B$ value by substituting $A=\\simplify[fractionnumbers]{{xcans}}$ into equation $(3)$:</p>\n<table>\n<tbody>\n<tr>\n<td>$B$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify[unitdenominator,!collectnumbers,fractionnumbers]{({-m}*({xcans})+{-p})/({n})}}$</td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>&nbsp;</td>\n</tr>\n<tr>\n<td></td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[fractionnumbers]{{ycans}}$</td>\n</tr>\n</tbody>\n</table>\n<p>Therefore the values that satisfy equations $(1)$ and $(2)$ are $A=\\simplify[fractionnumbers]{{xcans}}$ and $B=\\simplify[fractionnumbers]{{ycans}}$.</p>", "rulesets": {}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Polynomials: Equating coefficients 2 (simultaneous equations) [L4 Randomised]", "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Paul Hancock", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1738/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Paul Hancock", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1738/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}