// Numbas version: exam_results_page_options {"name": "Collecting like terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g"], "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "
Simplify the following by collecting like terms.
", "variables": {"b": {"templateType": "anything", "description": "", "definition": "shuffle(list(-10..10)+[0,0])[0..3]", "group": "Ungrouped variables", "name": "b"}, "f": {"templateType": "anything", "description": "", "definition": "shuffle(list(-10..10)+[0,0])[0..2]", "group": "Ungrouped variables", "name": "f"}, "d": {"templateType": "anything", "description": "", "definition": "shuffle(list(-10..10))[0..2]", "group": "Ungrouped variables", "name": "d"}, "g": {"templateType": "anything", "description": "", "definition": "shuffle(list(-10..10))[0..2]", "group": "Ungrouped variables", "name": "g"}, "a": {"templateType": "anything", "description": "", "definition": "shuffle(list(-10..10))[0..3]", "group": "Ungrouped variables", "name": "a"}, "c": {"templateType": "anything", "description": "", "definition": "shuffle(list(-10..10)+[0,0,0,0])[0..3]", "group": "Ungrouped variables", "name": "c"}}, "tags": ["algebra", "collecting like terms", "like terms", "simplifying"], "parts": [{"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "prompt": "$\\simplify[!collectnumbers]{{a[1]}x+{b[1]}y+{c[1]}z+{b[2]}y+{a[2]}x+{c[2]}z+{a[0]}x+{c[0]}z+{b[0]}y}$ = [[0]]
", "stepsPenalty": "1", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "steps": [{"customName": "", "customMarkingAlgorithm": "", "scripts": {}, "prompt": "Like terms are terms where the variable part is the same. For example, $4x$ and $-x$ have the same variable part $x$. However, $3x$ and $-2y$ have different variable parts and are therefore unlike terms (or not like terms).
\n\nWe can only collect like terms! Just like we can't say 2 m + 3 cm equals 5 m or 5 cm, we can't say $2x+3y$ equals $5x$ or $5y$! We can, however, say $2a+3a=5a$.
\n\nIn our question we look at all the terms with a variable part of $x$ and add up all the corresponding coefficients, we do the same for the $y$ terms and the $z$ terms:
\n\\[\\simplify[!collectnumbers]{{a[1]}x+{b[1]}y+{c[1]}z+{b[2]}y+{a[2]}x+{c[2]}z+{a[0]}x+{c[0]}z+{b[0]}y}=\\simplify[basic]{({a[1]}+{a[2]}+{a[0]})x+({b[1]}+{b[2]}+{b[0]})y+({c[1]}+{c[2]}+{c[0]})z}\\]
\nWe present this as the sum of three unlike terms:
\n\\[\\simplify{{sum(a)}x+{sum(b)}y+{sum(c)}z}\\]
", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "information", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "variableReplacements": []}], "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "scripts": {}, "answer": "{sum(a)}*x+{sum(b)}*y+{sum(c)}*z", "valuegenerators": [{"value": "", "name": "x"}, {"value": "", "name": "y"}, {"value": "", "name": "z"}], "extendBaseMarkingAlgorithm": true, "maxlength": {"message": "", "length": "14", "partialCredit": 0}, "variableReplacementStrategy": "originalfirst", "type": "jme", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}], "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "prompt": "$\\simplify[!collectnumbers]{{d[1]}x^2+{f[1]}x+{g[1]}+{d[0]}x^2+{f[0]}x+{g[0]}}$ = [[0]]
", "stepsPenalty": "1", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "steps": [{"customName": "", "customMarkingAlgorithm": "", "scripts": {}, "prompt": "Like terms are terms where the variable part is the same. For example, $4x$ and $-x$ have the same variable part $x$. However, $3x^2$ and $-2x$ have different variable parts and are therefore unlike terms (or not like terms).
\n\nWe can only collect like terms! Just like we can't say 2 m + 3 cm equals 5 m or 5 cm, we can't say $2x^2+3x$ equals $5x^2$ or $5x$! We can, however, say $2x^2+3x^2=5x^2$.
\n\nIn our question we look at all the terms with a variable part of $x^2$ and add up all the corresponding coefficients (the numbers in front of the variables), we do the same for the $x$ terms and the constant terms (the terms with no variable part):
\n\\[\\simplify[!collectnumbers]{{d[1]}x^2+{f[1]}x+{g[1]}+{d[0]}x^2+{f[0]}x+{g[0]}}=\\simplify[basic]{({d[1]}+{d[0]})x^2+({f[1]}+{f[0]})x+({g[1]}+{g[0]})}\\]
\nWe present this as the sum of three unlike terms:
\n\\[\\simplify{{sum(d)}x^2+{sum(f)}x+{sum(g)}}\\]
", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "information", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "variableReplacements": []}], "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "scripts": {}, "answer": "{sum(d)}x^2+{sum(f)}x+{sum(g)}", "valuegenerators": [{"value": "", "name": "x"}], "extendBaseMarkingAlgorithm": true, "maxlength": {"message": "", "length": "16", "partialCredit": 0}, "variableReplacementStrategy": "originalfirst", "type": "jme", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}], "variableReplacements": []}], "rulesets": {}, "advice": "", "name": "Collecting like terms", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}