4 x + 12 = a x + 3 2 In the given equation, a is a constant. If the

Question

4 x + 12 = 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 ?

Accepted Answer

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 = 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 = a x + 3 2 , has infinitely many solutions, then both sides of this equation are equal for any value of x . If x = 0 , then substituting 0 for x in the given equation yields 4 0 + 12 = a 0 + 3 2 , or 12 = 3 2 a . Dividing both sides of this equation by 3 2 yields 8 = 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 = 3 x + 3 2 , or 4 x + 12 = 3 2 x + 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 = 12 x + 3 2 , or 4 x + 12 = 6 x + 18 , which has one solution, not infinitely many solutions.

Question 162bcb