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Algebra / Systems of two linear equations in two variables Difficulty: Hard

$a x + b y = 72$

$6 x + 2 b y = 56$

In the given system of equations, $a$ and $b$ are constants. The graphs of these equations in the xy-plane intersect at the point $left parenthesis 4 comma y right parenthesis$ . What is the value of $a$ ?

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Explanation

Choice D is correct. It’s given that the graphs of the given system of equations intersect at the point $left parenthesis 4 comma y right parenthesis$ . Therefore, $left parenthesis 4 comma y right parenthesis$ is the solution to the given system. Multiplying the first equation in the given system by $negative 2$ yields $- 2 a x - 2 b y = - 144$ . Adding this equation to the second equation in the system yields $(- 2 a + 6) x + (- 2 b + 2 b) y = (- 144 + 56)$ , or $(- 2 a + 6) x = - 88$ . Since $left parenthesis 4 comma y right parenthesis$ is the solution to the system, the value of $a$ can be found by substituting $4$ for $x$ in this equation, which yields $(- 2 a + 6) (4) = - 88$ . Dividing both sides of this equation by $4$ yields $- 2 a + 6 = - 22$ . Subtracting $6$ from both sides of this equation yields $- 2 a = - 28$ . Dividing both sides of this equation by $negative 2$ yields $a equals 14$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.