ACT Math: Solving for One Variable in Terms of Another

In solving for variables there are two important things you need to remember.

  • Put like terms on one side

    If there is a particular variable you need to solve for, try to isolate that on one side of the equal sign and the other terms on the other side of the equal sign.

  • Performing Operations

    Any operation that you do to one side of the equation (adding, subtracting, dividing, multiplying), you also have to do to the other side of the equation.

Like terms means grouping the x’s together, the y’s together, the z’s together, and the real numbers together. The best way to learn is through examples, so here are a few question types to remember.

Basic Example

What is y in terms of x?

Suppose 4x + y = 2x + 2, what is y in terms of x?
In this case, we want to find y. So let’s leave y on the left hand side, and bring 4x over to the right hand side. Do this by subtracting 4x from both sides of the equation.
4x + y – 4x = 2x + 2 – 4x
y = 2 – 2x

For something involving division and multiplication, try this.

If y=(4(x+3))/b, then x+3 must equal?
The question wants x+3. So multiply both sides of the equation by b to get yb = 4(x+3)
Then divide both sides of the equation to get yb/4 = x+3

What if you have to deal with roots and squares? The same concepts applies.

For example, find a from 4√a  + 12  = 24
* subtract 12 from both sides of the equation: 4√a  = 12
* divide both sides by 4: √a = 3
* square both sides: a = 9


If you found the stuff on top easy, good for you. Here’s another type of algebraic manipulation.

Suppose 45% of a = b% of 90. What is a/b?

Setting up the equation and solving it,

A common mistake students often make with this question is to see the 90 and 45 and assume that a/b as a fraction must be ½. Always work out the equation to the end.

Relating Equations

Sometimes, you might not have to solve for x or y first. Don’t always be in a hurry to cross multiply or isolate x. Look at the equations you’re given and you may find that a complicated equation is perhaps the product or the addition of 2 other equations.
Here’s a simple example:

Given that y + 4 = k, what is the value of 2y + 8?
2y + 8 is 2 times of y + 4. So 2y + 8 = 2k

You don’t have to find y = k – 4 and then plug that into 2y + 8. Don’t do more work than you have to.


The last type of question deals with exponents. Whenever you see exponents, you generally want to try and make everything in the equation have the same base and then compare the exponents.
Here’s what I mean.

If  23+x = 82x-4, what is x?
Since 8 = 23 Then 23+x = (23)2x-4 = 26x-12
Now that the bases are the same – its in base 2. We can look at the powers and compare them.
So 3 + x = 6x – 12. From here, solving for x is pretty simple. (Did you get x = 3?)