Digital PSAT Math Practice Question 344: Answer and Explanation

Question: 344

Consider the following system of equations with variables A and B and constant integers X and Y:

By what number must the sum of X and Y be divisible in order for the two equations to have infinitely many solutions?

Grid-In:

Correct Answer: 3

Explanation:

3 There will be infinitely many solutions if the two equations are multiples of the same equation. The coefficients of the A and B terms in $A+2B=4$ add up to 3 because they are 1 and 2. Since $XA+YB=4X$ is divisible by 4 on the right-hand side, as is the other equation, the sum of X and Y must also be divisible by 3 in order for the two equations to be multiples of one another. To replicate the structure of the first equation, Y must equal 2X so that the two equations will be multiples of one another. To see this with greater clarity, consider this example:

If the second equation had $X=2$ and $Y=4$, the equation would be twice the first equation: $2A+4B=8$. This equation is simply a multiple of the first one, making them essentially identical. As a result, there are infinitely many solutions since the equations overlap each other when graphed.