Multiple choice

Solve the following systems of equations using elimination method. $\dfrac{5}{x}-\dfrac{4}{y}=-2, \dfrac{2}{x}+\dfrac{3}{y}=13$

  1. $\left ( \dfrac{1}{2}, \dfrac{1}{3} \right )$
  2. $\left ( \dfrac{1}{3}, \dfrac{1}{3} \right )$
  3. $\left ( \dfrac{1}{2}, \dfrac{1}{4} \right )$

  4. None of these
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A Correct answer
Explanation

Substituting x=1/2 and y=1/3 into the first equation: 5/(1/2) - 4/(1/3) = 10 - 12 = -2. Substituting into the second: 2/(1/2) + 3/(1/3) = 4 + 9 = 13. Both equations are satisfied.

AI explanation

Using the substitution method, let 1/x equal u and 1/y equal v, which changes the equations to 5u - 4v equals -2 and 2u + 3v equals 13. Multiplying the first equation by 3 gives 15u - 12v equals -6, and multiplying the second by 4 gives 8u + 12v equals 52. Adding these resulting equations eliminates v and yields 23u equals 46, so u equals 2; substituting this back gives 2(2) + 3v equals 13, meaning v equals 3. Since u is 1/x and v is 1/y, the values of x and y are 1/2 and 1/3 respectively.

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