Some GRE problems, including word problems, give you two equations, each of which has two variables. To solve such problems, you’ll need to solve for one or each of those variables. At first glance, this problem may seem quite daunting:
If 3x + y = 10 and y = x − 2, what is the value of y?
Maybe you’ve gotten pretty good at solving for one variable, but now you face two variables and two equations.
You might be tempted to test numbers, and indeed you could actually solve this problem that way. Could you do so in under two minutes? Maybe not. Fortunately, there is a much faster way.
One method for combining equations is called substitution. In substitution, you insert the expression for one variable in one equation into that variable in the other equation. The goal is to end up with one equation with one variable, because once you get a problem to that point, you know you can solve it.
There are four basic steps to substitution, which can be demonstrated with the previous question.
Step One is to isolate one of the variables in one of the equations. For this example, y is already isolated in the second equation: y = x − 2.
For Step Two, it is important to understand that the left and right sides of the equation are equivalent. This may sound obvious, but it has some interesting implications. If y equals x − 2, then that means you could replace the variable y with the expression (x − 2) and the equation would have the same value. In fact, that’s exactly what you’re going to do. Step Two will be to go to the first equation and replace the variable y with its equivalent, (x − 2). In other words, you’re substituting (x − 2) in for y. So:
3x + y = 10 → 3x + (x − 2) = 10
Now for Step Three, you have one equation and one variable, so the next step is to solve for x:
Now that you have a value for x, Step Four is to substitute that value into either original equation to solve for your second variable, y:
y = x − 2 → y = 3 − 2 = 1
So the answer to the question is y = 1. It should be noted that Step Four will only be necessary if the variable you solve for in Step Three is not the variable the question asks for. The question asked for y, but you found x, so Step Four was needed to answer the question.
Now that you’ve gotten the hang of substitution, try a new problem:
If 2x + 4y = 14 and x − y = −8, what is the value of x?
As you learned, the first step is to isolate your variable. Because the question asks for x, you should manipulate the second equation to isolate y. Taking this approach will make Step Four unnecessary and save you time:
Then, for Step Two, you can substitute for y in the first equation:
Now, for Step Three, isolate x:
So the answer to the question is x = −3.
Solve for x and y in the following equations.
x = 10
x + 2y = 26
x + 4y = 10
y − x = −5
6y + 15 = 3x
x + y = 14