As you can see, those two equations form parallel lines with different y
intercepts. Because they never intersect, there is no coordinate pair (x,y) that will
satisfy both equations.
Similarly, you might get two equations in which the solution is all real
numbers. Like so:
2 y + 13x = 12
36 – 39x = 6yAny pair of x,y coordinates that works for the first equation will work for the
second. This is because the second equation is exactly the same as the first, just
multiplied by 3 and with its terms shifted around to different sides of the equals
sign. The SAT likes to play games like this, so be on the lookout.
Absolute Value
Absolute value problems also work very similarly to standard algebraic
equations, with one key difference: They break into two equations. Here’s what
we mean.
Why do we do this? Because whether the expression “5x + 6” comes out to
66 or –66, the absolute value sign is going to make it positive 66, so both would
work. Now, we just solve both equations in the regular way.
So the two possible values for x are 12 and –72 / 5.Samantha’s Theorem*: | –x | = | × | = x
*(Okay, this isn’t really a theorem. But it is my proprietary
Math Fact, and I will sue if you copy it.)