Cracking The SAT Premium

gx  –   hy  =   78

4 x +   3y  =   13

In  the system  of  equations   above,  g   and h   are constants.  If  the system  has infinitely  many

solutions,  what    is  the value   of  gh  ?

A) –432

B) –6

C) 6

D) 432

To  Infinity...and  Beyond!

When    given   two equations

with    infinitely  many    solu-

tions,  find    a   way to  make

them    equal.  The equations

represent   the same    line.

Here’s How to Crack It

This question may have you scratching your head and moving on to the next question, but let’s explore
what you can do to solve this before you decide it’s not worth your time. You may be surprised by how
easy it is to solve a problem like this.

When they say that these equations have infinitely many solutions, what they are really saying is that these
are the same equation, or that one equation is a multiple of the other equation. In other words, these two
equations represent the same line. With that in mind, try to determine what needs to be done to make these
equations equal. Since the right side of the equation is dealing with only a constant, first determine what
you would need to do to make 13 equal to 78.

In this case, you need to multiply 13 by 6. Since we are working with equations, we need to do the same
thing to both sides of the equation in order for the equation to remain equal.

6(4x    +   3y) =   6   ×   13

24 x    +   18y =   78

Since both equations are now equal to 78, you can set them equal to one another, giving you this equation:

24 x    +   18y =   gx  –   hy

You may know that when you have equations with the same variables on each side the coefficients on
those variables must be equal, so you can deduce that g = 24 and h = –18. (Be cautious when you evaluate
this equation. The test writers are being sneaky by using addition in one equation and subtraction in
another.) Therefore, gh equals 24 × –18 = –432. Choice (A) is correct.