Isolating variables and expressions

Isolating variables

In order to solve most algebra problems it is almost always essential to begin by getting one variable all alone on one side of the equation. We do this because it allows us to say that a variable is equal to the remainder of the equation. For instance, in the equation $y=2x+16$ we can find what $y$ is equal to by solving for everything else in the equation, $2x+16$.

Sometimes, we are given equations that do not already have the key variable isolated. This could look like $2x-y=16$. In this equation, we can no longer solve so easily for a single variable; we will have to rearrange it and isolate the variable that we want to solve for. Let’s figure out how to do that.

PEMDAS

To isolate a variable, we will need to use a reverse order of operations to play around with the terms in the equation. Normally we use PEMDAS to simplify an equation. In this section, we will use PEMDAS backward in order to reverse engineer the question and make it do what we want to do. You could also think of this method as working from the outermost parts of the equation in relation to $x$ to the innermost parts in relation to $x$, which is opposite to what we normally do to simplify equations with PEMDAS.

In short, we need to perform opposite operations:

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

We will use this equation as we learn how to isolate variables:

$$4x+2=10$$

This is a simple equation with one variable. The purpose of isolating $x$ in this equation is to find the exact value of $x$. Let’s start by figuring out how to get $x$ by itself on the left side of the equation. There are two things we need to do before we can accomplish this:

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

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

Which of these should we do first? Following reverse PEMDAS, or starting from the number furthest away from $x$, we have to deal with the addition of $2$ first. We can deal with addition by simply subtracting the number from both sides of the equation. Adding this action to the equation, we get

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

Look what happens now. The left side of the equation has $2-2$, which is equal to $0$. In other words, the equation now looks like

$$4x=8$$

We have now successfully eliminated the added $2$. This demonstrates the golden rule of algebraic operations: What you do to one side of the equation, you must do to the other side as well. So, if you need to subtract $2$ from the left side of the equation to get $x$ alone, you have to do the same to the right side.

Now, we have to get rid of the $4$ being multiplied by $x$. What do we need to do to cause the $4$ to be canceled out? We need to divide both sides by $4$.

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

Now simplifying the fractions, our final equation is

$$x=2$$

This is the purpose of isolating variables! We get to clearly identify the value of $x$ without having to do any guess work.

Examples

Try some examples on your own to get a hang of this. It might be difficult at the start, but once you build the mental muscle for it you’ll be able to crush it in no time.

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 we know how to play with equations using algebra, we can look at how to evaluate the functions that we create! A function has two variables, and is used to solve for the value of one of the variables as we change the other. For instance, a simple function is $y=x+1$. As we increase the value of $x$, the value of $y$ also increases. This is a function for $y$.

All we have to do when we evaluate functions is take a number we are given and put it in the place of a variable within the equation. Let’s do an example to make sense of this. We will use the same equation as above.

Evaluate the following function when $x=3$

$y=x+1$

What’s next? If the value of $x$ is $3$, then we can replace $x$ with $3$ in the equation so that it looks like:

$$y=3+1$$

This part is easy. Just simplify the right side of the equation to figure out what $y$ equals: $4$.

Isolating and evaluating

Sometimes you will be required to first isolate a variable, then evaluate the function that it creates. For example:

Evaluate the following function when $x=2$

$2x-y=4$

First, isolate the variable we need to find ($y$) by adding $y$ to both sides and subtracting $4$ from both sides:

$$2x-4=y$$

Now, replace the variable we are given ($x$) with the number it is equal to ($2$):

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

Our final answer to this problem is $y=0$.

Function notation

When a question wants you to evaluate a function, oftentimes you will see the notation “$f(x)$” instead of $y$. This is practically saying “function using $x$ as the variable.” So, the example above could be rewritten as

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

This looks a bit different, but you could imagine it asking “evaluate the function using $3$ as the variable.” So, we use the same process. Place a $3$ where the $x$ is in the equation, and simplify it to find the answer $f(3)=4$.

Try another example.

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

Layered functions

We’re going to level up in difficulty in this next section, so make sure you have a good handle on the previous section before moving on to this one.

A layered function is one that uses multiple functions at one time. An example looks like $f(g(x))$. This is also sometimes called the “Composition of Functions.” You can see that the example begins with the usual “function using…” but then a new function shows up. So, we will replace that variable in the equation with this new function $g(x)$. Let’s examine what this looks like:

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

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

$$g(x)=3$$

Let’s solve this in the order that we read it. Write out the function $f$ since it comes first:

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

Now, instead of using $x$ we will use $g(x)$. Let’s make that change:

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

We simply put the whole function of $g$ where $x$ would normally be. Now, what should we do with $g(x)$? We cannot leave that in the equation. We know from the problem that $g(x)$ is equal to $3$, so we will replace $g(x)$ with $3$:

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

This is our final answer! Just simplify to get the answer $f(g(x))=7$.

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.