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Algebra / Linear equations in one variable Difficulty: Medium

$4 x + 12 = \frac{a (x + 3)}{2}$

In the given equation, $a$ is a constant. If the equation has infinitely many solutions, what is the value of $a$ ?

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Explanation

Choice C is correct. If an equation has infinitely many solutions, then the two sides of the equation must be equivalent. Multiplying each side of the given equation by $2$ yields $8 x + 24 = a (x + 3)$ . Since $8$ is a common factor of both terms on the left-hand side of this equation, the equation can be rewritten as $8 (x + 3) = a (x + 3)$ . The two sides of this equation are equivalent when $a equals 8$ . Therefore, if the given equation has infinitely many solutions, the value of $a$ is $8$ .

Alternate approach: If the given equation, $4 x + 12 = \frac{a (x + 3)}{2}$ , has infinitely many solutions, then both sides of this equation are equal for any value of $x$ . If $x equals 0$ , then substituting $0$ for $x$ in the given equation yields $4 (0) + 12 = \frac{a (0 + 3)}{2}$ , or $12 equals three halves a$ . Dividing both sides of this equation by $\frac{3}{2}$ yields $8 equals a$ .

Choice A is incorrect. If the value of $a$ is $0$ , the given equation is equivalent to $4 x + 12 = 0$ , which has one solution, not infinitely many solutions.

Choice B is incorrect. If the value of $a$ is $3$ , the given equation is equivalent to $4 x + 12 = \frac{3 (x + 3)}{2}$ , or $4 x + 12 = \frac{3}{2} x + \frac{9}{2}$ , which has one solution, not infinitely many solutions.

Choice D is incorrect. If the value of $a$ is $12$ , the given equation is equivalent to $4 x + 12 = \frac{12 (x + 3)}{2}$ , or $4 x + 12 = 6 x + 18$ , which has one solution, not infinitely many solutions.