PSAT Math Grid-Ins Question 37: Answer and Explanation

Question: 37

y = x + 8

y = x² - x - 7

If (x, y) is a solution to the equations above, what is one possible value for the sum of x and y?

Correct Answer: 2 or 18

Explanation:

2 or 18

The question asks for the sum of x and y and provides a system of equations. Since both equations are set equal to the same value, setting the right sides of the two equations equal will be the best method to solve the system for x. The new equation becomes x + 8 = x² - x - 7. Because this is a quadratic equation, get one side equal to 0. Subtract 8 from both sides to get x = x² - x - 15, and subtract x from both sides to get 0 = x² - 2x - 15. Factor the right side. Find two numbers with a product of -15 and a sum of -2. These are 3 and -5. Therefore, the equation can be written as 0 = (x + 3)(x - 5). Set each factor equal to 0 to get x + 3 = 0 and x - 5 = 0. The question asks for one possible sum of x and y, so it is necessary to get only one value of x. Therefore, one possible solution is to subtract 3 from both sides of x + 3 = 0 to get x = - 3. Plug this value of x into the first equation to get y = -3 + 8 = 5. Therefore, one possible sum of x and y is -3 + 5 = 2. Alternatively, to get the other possible solution, add 5 to both sides of x - 5 = 0 to get x = 5. Plug this value of x into the first equation to get y =5 + 8 = 13. Therefore, the other possible sum of x and y is 5 + 13 = 18. The correct answer is either 2 or 18.