Algebra practice question
In the system of equations 2x + 3y = 12 and 5x − 2y = k, where k is a constant, the solution (x, y) satisfies x = y. What is the value of k?
Answer: 7.2 or 36/5
Because the solution satisfies x = y, substitute y = x into the first equation: 2x + 3x = 12, so 5x = 12 and x = 12/5. Then y = 12/5 as well, and k = 5x − 2y = 5(12/5) − 2(12/5) = 12 − 24/5 = 36/5 = 7.2. Common wrong answers: 16.8 (= 84/5) comes from a sign error, evaluating 5x + 2y instead of 5x − 2y; 6 comes from rounding x = 2.4 down to the integer 2 before substituting (5·2 − 2·2 = 6); 2.4 comes from stopping after finding x rather than evaluating the second expression; 3 comes from mistakenly treating k as 5 − 2 times a solution value of 1.
Common wrong answers
- If you answered 3
- The second equation is k = 5x − 2y with x = y = 12/5, not 5 − 2; substituting gives k = 7.2.
- If you answered 6
- You rounded x = 2.4 down to 2 before substituting. Keep the exact value 12/5: k = 5(12/5) − 2(12/5) = 7.2.
- If you answered 16.8
- You evaluated 5x + 2y instead of 5x − 2y; with x = y = 12/5 the correct value is 12 − 24/5 = 7.2.
- If you answered 2.4
- That's x (and y), but k is the value of 5x − 2y, which comes out to 7.2.
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