Simultaneous equations question. values for the coefficients are generated to be small numbers, random values are generated for the weights and the resultant energies are calculated for the question. Student needs to solve equations to find coefficients. Advice gives solution using method of elimination.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "Using the available data we have \\begin{align} \\var{E_1}&=\\var{W_1}a+b\\quad (1)\\\\ \\var{E_2}&=\\var{W_2}a+b\\quad (2)\\end{align}
\nUsing the method of elimination to eliminate \\(b\\) subtract equation \\((1)\\) from equation \\((2)\\) to get \\[\\simplify{{E_2-E_1}}=\\simplify{{W_2-W_1}}a\\]
\nand hence \\[a=\\var{a}\\]
\nSubstituting this into \\((1)\\) gives us \\[\\var{E_1}=\\simplify{{W_1*a}}+b\\] and hence \\begin{align}\\var{E_1}&=\\simplify{{W_1*a}}+b\\\\b&=\\var{E_1}-\\simplify{{W_1*a}}\\\\&=\\var{b}\\end{align}
\nThus the constants are \\(a=\\var{a}\\) and \\(b=\\var{b}\\).
\n(Note that this problem could also be solved by the method of substitution, however the final answers will be the same.)
", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(0.1..0.9 #0.1)", "description": "value of coefficient for lifting equation - random 0.1 to 0.9 in steps of 0.1. Note that this one of the solutions that the student has to find.
", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(5..20)", "description": "Value of constant term in lifting equation - random integer between 5 and 10. This is the other solution for the question.
", "templateType": "anything", "can_override": false}, "W_1": {"name": "W_1", "group": "Ungrouped variables", "definition": "random(70..200 #10)", "description": "Randomly generated value for first weight
", "templateType": "anything", "can_override": false}, "W_2": {"name": "W_2", "group": "Ungrouped variables", "definition": "random(80..250 #10 except W_1)", "description": "Randomly generated value for second weight, cannot be the same as W_1.
", "templateType": "anything", "can_override": false}, "E_1": {"name": "E_1", "group": "Ungrouped variables", "definition": "a*W_1+b", "description": "Resulting energy required to lift weight 1
", "templateType": "anything", "can_override": false}, "E_2": {"name": "E_2", "group": "Ungrouped variables", "definition": "a*W_2+b", "description": "Resulting energy required to lift weight 2.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "W_1", "W_2", "E_1", "E_2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "A lifting machine obeys the law \\[E=aW+b\\]where \\(E\\) is effort force, \\(W\\) is load and \\(a\\) and \\(b\\) are constants. An experiment produces the following results: An effort force of \\(\\var{E_1}\\) N lifts a load of \\(\\var{W_1}\\) N while an effort force of \\(\\var{E_2}\\) N lifts a load of \\(\\var{W_2}\\) N. Find the values of the constants \\(a\\) and \\(b\\). Write your final answer in the box below and show full working on your handwritten notes. (Note: Leave answers in fraction form where appropriate. For example to enter the fraction \\(\\frac32\\) type 3/2.)
\\(a=\\)[[0]]
\n\\(b=\\)[[1]]
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