Multiple choice

Solve the following equation. $2^x+2^y=20, x+y=6$.

  1. x=2,y=4

  2. x=3,y=3

  3. x=3,y=2

  4. x=0,y=2
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A Correct answer
Explanation

Given x+y=6, we can substitute y=6-x into the first equation to get 2^x + 2^(6-x) = 20. Multiplying by 2^x gives (2^x)^2 - 20(2^x) + 64 = 0, which factors to (2^x - 16)(2^x - 4) = 0. This yields 2^x=16 (x=4, y=2) or 2^x=4 (x=2, y=4).

AI explanation

From the linear equation x + y = 6, we can express y as 6 - x and substitute this into the first equation to get 2^x + 2^(6 - x) = 20. Multiplying the entire equation by 2^x yields a quadratic in terms of 2^x, specifically (2^x)^2 - 20(2^x) + 64 = 0. Factoring this quadratic gives (2^x - 4)(2^x - 16) = 0, meaning 2^x is either 4 or 16. Solving these gives x = 2 or x = 4, which correspond to y = 4 or y = 2 respectively, making x = 2 and y = 4 a valid solution.

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