3.3 SAT Math - Substitution

If you need to solve for two variables, you can substitute information you know about one variable to help solve a second equation.

Substitution Example

3. If x = 15 and y = 10, what is the value of 4x(x - y)?

In this case, substitute in your values for x and y to get the number they’re looking for:

4•15(15 - 10) = 300

3. If x = 3, what is y in the equation 3y - x = 12?

In this case, substitute 3 for x and solve for y.

3y -3 = 12
3y = 15
y = 5

Example Problem 2

If you’re not given any numbers, but you are given more than one equation on the SAT exam, pick one equation to start with, solve for one variable in that equation, and then substitute what you’ve found into the second equation:

7. If 3x = 9and 7x +2y = 25, what is y?

Pick the easier equation to solve for first.  In this case, that’s 3x = 9.  Divide by 3 on both sides to get x = 3, and then plug that number into the second equation:

7•3 + 2y =25
21 + 2y = 25
2y = 4
y = 2

You must always have at least one equation for every variable that you need to solve!  Sometimes that equation can be as simple as x = 3or it can be more complicated.  In more complicated versions, the most you will be able to do to the first equation is to put it in terms of the other; that is, to clean it up so only one variable is on one side, and everything else is on the other.  Then, substitute that whole thing in for the variable in the second equation.

12. If 2x = y + 3 and 3y - 3x = 6, find x and y.

1: You always want to work with the simpler equation first, but in this case, they’re both relatively equal in difficulty.  Pick the simpler variable to solve for first.   We’re going to use y and the first equation.

2: Get y all by itself on one side of the equation.

2x - 3 = y

3: Anywhere you see a y in the second equation, replace it with this new information.

3y - 3x = 6
3(2x - 3) - 3x = 6

4: Now you are only dealing with one variable in your equation, and can solve for x.

3(2x - 3) - 3x = 6
6x - 9 - 3x = 6
3x = 15
x = 5

5: Now that you know what number x equals, replace the x in your first equation (from Step 2) with that number and solve for y.

2x - 3 = y
2•5 - 3 = y
7 = y

So x = 5 and y = 7.

This way of solving for two variables is called substitution.  Sometimes you can use another way of solving called elimination.