2 5 x + 7 5 y = 2 7 g x + k y = 5 2 In the

Question

2 5 x + 7 5 y = 2 7 g x + k y = 5 2 In the given system of equations, g and k are constants. The system has infinitely many solutions. What is the value of g k ?

Accepted Answer

The correct answer is 2 7 . It&rsquo;s given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions if and only if the two linear equations are equivalent. Multiplying each side of the first equation in the system by 35 4 yields 35 4 2 5 x + 7 5 y = 35 4 2 7 , or 7 2 x + 49 4 y = 5 2 . Since this equation is equivalent to the second equation and has the same right side as the second equation, the coefficients of x and y , respectively, should also be the same. It follows that g = 7 2 and k = 49 4 . Therefore, the value of g k is 7 2 49 4 , or 2 7 . Note that 2/7, .2857, 0.285, and 0.286 are examples of ways to enter a correct answer.

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