PSAT Math Multiple-Choice Question 687: Answer and Explanation

Question: 687

Out of all possible solutions (x, y) to the pair of equations below, what is the greatest possible product xy that can be obtained?

x(y + 2) - 3x - 4(y + 2) = -12
and
3x - 6 = 3y

• A. 3
• B. 4
• C. 6
• D. 8

Correct Answer: D

Explanation:

(D) The second equation is simpler, so solve for y and then plug your expression for y back into the first equation:

3x - 6 = 3y

Divide by 3 to solve for y:

Now you can plug x - 2 in for y in the first equality:

x[(x - 2) + 2] - 3x - 4[(x -2) + 2] = -12

Combine like terms within the parentheses:

x(x) - 3x - 4(x) = -12

Combine like terms and bring the 12 to the left side:

x² - 7x + 12 = 0

Factor this, or use the quadratic equation if you're not great at factoring:

(x - 3)(x - 4) = 0

Set each factor equal to 0 to solve for the possible values of x:

x - 3 = 0
x = 3
x - 4 = 0
x = 4

Next, plug these values into either of the two equations to solve for y. It'll be easiest to plug them into the equation that you already solved for y:

y = x - 2 = 3 - 2 = 1

y = x - 2 = 4 - 2 = 2

So x can equal 3 or 4, and y can equal 1 or 2. Therefore, the greatest product xy will be:

4 × 2 = 8

Choice (D) is the answer.