// Numbas version: finer_feedback_settings {"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": ""}}, "statement": "

Suppose that during some working, you find that

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\\[\\simplify{{-s}x+{-p}={q}*A*x+{r}*B*x+{m}A+{n}B}\\]

\n

for all $x$.

", "tags": ["l4", "L4"], "ungrouped_variables": [], "extensions": [], "variable_groups": [{"variables": ["m", "n", "p", "q", "r", "s", "xcans", "ycans"], "name": "part c"}], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": ""}, "name": "Polynomials: Equating coefficients 2 (simultaneous equations) [L4 Randomised]", "advice": "

Two polynomials are equal if and only if their corresponding coefficients are also equal.

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So we 'equate the coefficients' of the polynomials given in this question, this means that the constant terms must be equal, ie

\n

\n

\\[\\simplify{{-p}={m}*A+{n}*B}\\qquad\\qquad(1)\\]

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the coefficients of the $x$ terms must be equal, ie

\n

\\[\\simplify{{-s}={q}A+{r}*B}\\qquad\\qquad(2)\\]

\n

\n

We now solve these equations simultaneously and find $A=\\simplify[fractionnumbers]{{xcans}}$ and $B=\\simplify[fractionnumbers]{{ycans}}$.

\n

\n
\n

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The working for simulations equations is given below.

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There are many ways to solve these equations simultaneously. Here is one method.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{m}A+{n}B}$$=$$\\var{-p}$               $(1)$
$\\simplify{{q}A+{r}B}$$=$$\\var{-s}$               $(2)$
\n

Solve one of the equations for one of the variables. Here we solve equation $(1)$ for $B$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{n}B$$=$$\\simplify{{-m}A+{-p}}$
 
$B$$=$$\\displaystyle{\\simplify{({-m}A+{-p})/({n})}}$               $(3)$
\n

Substitute this expression for $B$ given in $(3)$ into $(2)$:

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\\[\\simplify[all,!collectnumbers]{{q}A+{r}*(({-m}A+{-p})/{n}) =- {s}}\\]

\n

Collect like terms:
\\[\\simplify[fractionnumbers]{{q-r*m/n}A={-s+r*p/n}}\\]

\n

Solve for $A$:
\\[A=\\simplify[fractionnumbers]{{xcans}}\\]

Now we know the $A$ value we can determine the corresponding $B$ value by substituting $A=\\simplify[fractionnumbers]{{xcans}}$ into equation $(3)$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$B$$=$$\\displaystyle{\\simplify[unitdenominator,!collectnumbers,fractionnumbers]{({-m}*({xcans})+{-p})/({n})}}$
 
$=$$\\simplify[fractionnumbers]{{ycans}}$
\n

Therefore the values that satisfy equations $(1)$ and $(2)$ are $A=\\simplify[fractionnumbers]{{xcans}}$ and $B=\\simplify[fractionnumbers]{{ycans}}$.

", "preamble": {"js": "", "css": ""}, "rulesets": {}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"showCorrectAnswer": true, "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "type": "gapfill", "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "prompt": "

By equating coefficients, determine the values of $A$ and $B$.

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$A=$ [[0]],   $B=$ [[1]]

\n

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