Multiple choice

Solve the following system of equation for x and y $\displaystyle \frac{2}{x}+\frac{3}{y}=2;\, \frac{8}{x}-\frac{6}{y}=2$

  1. $\displaystyle \frac{1}{2},\frac{1}{3}$
  2. $\displaystyle \frac{1}{3},\frac{1}{2}$

  3. 2, 3

  4. 3, 2
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C Correct answer
Explanation

Let u = 1/x and v = 1/y. The equations become 2u + 3v = 2 and 8u - 6v = 2. Multiplying the first by 2 gives 4u + 6v = 4, and adding this to the second gives 12u = 6, so u = 1/2 (x=2). Substituting back, 2(1/2) + 3v = 2 gives 3v = 1, so v = 1/3 (y=3).

AI explanation

By substituting u for 1 divided by x and v for 1 divided by y, the given equations become 2u plus 3v equals 2 and 8u minus 6v equals 2. Multiplying the first equation by 2 and adding it to the second equation eliminates v, yielding 12u equals 6, so u equals 1 divided by 2. Substituting u back gives v equals 1 divided by 3, and returning to the original variables gives x equals 2 and y equals 3.

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