PSAT Math Grid-Ins Question 41: Answer and Explanation

Question: 41

k(10 x - 5) = 2(3 + x) - 7

If the equation above has infinitely many solutions and k is a constant, what is the value of k ?

Correct Answer: 1/5 or .2

Explanation:

1/5 or.2

Difficulty: Medium

Category: Heart of Algebra/Linear Equations

Getting to the Answer: There are two variables but only one equation, so you can't actually solve the equation for k. Instead, recall that an equation has infinitely many solutions when the left side is identical to the right side. When this happens, everything cancels out and you get 0 = 0, which is always true. Start by simplifying the right-hand side of the equation. Don't simplify the left side because k is already in a good position:

Next, compare the left side of the equation to the right side. Rather than distributing the k, notice that 2 x is a fifth of 10 x and -1 is a fifth of -5, so if k were (or 0.2), then both sides of the equation would equal 2 x - 1, and it would therefore have infinitely many solutions. Thus, k is 1/5 or .2. Grid in either of these responses.