Isolating variables and expressions

Isolating variables

To solve most algebra problems, you’ll usually start by getting one variable by itself on one side of the equation. This is called isolating a variable. Once a variable is isolated, you can clearly see what it’s equal to.

For example, in the equation $y=2x+16$, the variable $y$ is already isolated. That means $y$ is equal to the entire expression $2x+16$.

Sometimes, the variable you want isn’t isolated. For example, in $2x-y=16$, neither variable is alone on one side. In that case, you need to rearrange the equation to isolate the variable you want.

PEMDAS

To isolate a variable, you use the order of operations in reverse. When you simplify expressions, you follow PEMDAS. When you isolate a variable, you “undo” operations in the opposite order.

A helpful way to think about this: start with the operation furthest from the variable and work inward.

In short, you use opposite operations:

Operation in the equation	What we do to get rid of it
Subtraction	Add
Addition	Subtract
Division	Multiply
Multiplication	Divide

We’ll use this equation to practice isolating a variable:

$$4x+2=10$$

This equation has one variable, $x$. The goal is to isolate $x$ so you can find its exact value.

To isolate $x$, you need to:

1. Get rid of the $2$ that is added to the $x$

2. Get rid of the $4$ that is multiplied by $x$

Which should you do first? Since $2$ is added after the multiplication, you undo the addition first. To undo “$+2$”, subtract $2$ from both sides of the equation:

$$4x+2-2=10-2$$

On the left side, $2-2=0$, so the equation becomes:

$$4x=8$$

This shows a key rule in algebra: Whatever you do to one side of the equation, you must do to the other side as well.

Now $x$ is still multiplied by $4$. To undo multiplication by $4$, divide both sides by $4$:

$$(4x)/4=8/4$$

Simplifying gives:

$$x=2$$

That’s the purpose of isolating variables: you can find the value of a variable directly, without guessing.

Examples

Try these examples to practice isolating variables.

What is the value of $x$ in the equation $2x-3=6$?

What is the value of $y$ in the equation $x/6-3=-2$?

Find an expression for $x$ in the following equation (you will not get a numerical answer)

$$x/y=2+4y$$

Evaluation of functions

Now that you know how to rearrange equations, you can evaluate functions.

A function is an equation that describes how one variable depends on another. For example, the function $y=x+1$ tells you that when $x$ increases by 1, $y$ also increases by 1.

To evaluate a function, you replace the variable with a given number and then simplify.

Evaluate the following function when $x=3$

$y=x+1$

If $x=3$, replace $x$ with $3$:

$$y=3+1$$

Now simplify:

$$y=4$$

Isolating and evaluating

Sometimes you’ll need to isolate a variable first, and then evaluate the resulting expression.

Evaluate the following function when $x=2$

$2x-y=4$

First, isolate the variable you need to find ($y$). Add $y$ to both sides and subtract $4$ from both sides:

$$2x-4=y$$

Now substitute $x=2$:

$$2(2)-4=y=0$$

Our final answer is $y=0$.

Function notation

When a problem asks you to evaluate a function, you’ll often see function notation like $f(x)$ instead of $y$.

• $f(x)$ means “the output of function $f$ when the input is $x$.”

So this example:

Evaluate $f(3)$ for the following function: $f(x)=x+1$

means: replace $x$ with $3$ and simplify.

• $f(3)=3+1=4$

Try another example.

Given the function $f(x)=(\frac{1}{2})x$, evaluate $f(12)$.

Layered functions

A layered function uses one function inside another. You might see this written as $f(g(x))$. This is also called the composition of functions.

The idea is: evaluate the inside function first, then use that result as the input to the outside function.

Evaluate $f(g(x))$ for the following functions:

$$f(x)=2x+1$$

$$g(x)=3$$

Start with the definition of $f$:

$$f(x)=2x+1$$

Now replace $x$ with $g(x)$:

$$f(g(x))=2(g(x))+1$$

Since $g(x)=3$, substitute $3$:

$$f(g(x))=2(3)+1$$

Simplify:

• $f(g(x))=7$

Let’s try another example.

Given:

$$f(x)=x-2$$

$$g(x)=1+2x$$

Evaluate the function $g(f(2))$.

Key points

Isolating variables. Move all unwanted numbers and variables to the other side of the equation by using opposite operations (addition is opposite subtraction, multiplication is opposite division).

Evaluating functions. Replace the variable in an equation with the value to which it is equal.

Function notation. $f(x)$ is sometimes used instead of $y$, and it effectively means “function $f$ using $x$ as the variable…”

Layered functions. When a function has more than one function at once, $f(g(x))$ for instance, write out the equations in the order that they are read. Begin with function $f(x)$, then replace $x$ with the function $g(x)$, then replace that $x$ with the value given.