Multiple choice

Solve the following equations by substitution method. $\dfrac{3}{x} + \dfrac{1}{y} = 7; \dfrac{5}{x} - \dfrac{4}{y} = 6. (x \neq 0, y \neq 0)$

  1. $x = \dfrac{1}{2}, y = 1$
  2. $x = \dfrac{-1}{2}, y = 1$
  3. $x = \dfrac{1}{2}, y = -1$

  4. None of these
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A Correct answer
Explanation

Let u = 1/x, v = 1/y. 3u + v = 7 and 5u - 4v = 6. Multiply first by 4: 12u + 4v = 28. Add to second: 17u = 34 => u = 2. Then x = 1/2. Substitute u=2 into 3u+v=7: 6+v=7 => v=1. Then y = 1.

AI explanation

Using the substitution method, let u equal 1/x and v equal 1/y to rewrite the system as 3u + v equals 7 and 5u - 4v equals 6. From the first equation, v equals 7 - 3u; substituting this into the second equation gives 5u - 4(7 - 3u) equals 6, which simplifies to 17u equals 34, so u equals 2. Substituting u back into the expression for v yields v equals 7 minus 6, which is 1. Since u equals 1/x and v equals 1/y, x equals 1/2 and y equals 1.

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