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Algebra / Linear equations in one variable Difficulty: Medium

$2 x + 16 = a (x + 8)$

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

The correct answer is $2$ . An equation with one variable, $x$ , has infinitely many solutions only when both sides of the equation are equal for any defined value of $x$ . It's given that $2 x + 16 = a (x + 8)$ , where $a$ is a constant. This equation can be rewritten as $2 (x + 8) = a (x + 8)$ . If this equation has infinitely many solutions, then both sides of this equation are equal for any defined value of $x$ . Both sides of this equation are equal for any defined value of $x$ when $2 equals a$ . Therefore, if the equation has infinitely many solutions, the value of $a$ is $2$ .

Alternate approach: If the given equation, $2 x + 16 = a (x + 8)$ , 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 $2 x + 16 = a (x + 8)$ yields $2 (0) + 16 = a (0 + 8)$ , or $16 equals 8 a$ . Dividing both sides of this equation by $8$ yields $2 equals a$ .