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1. ACT Math
1.1 ACT math intro
1.2 Pre-algebra
1.3 Elementary algebra
1.3.1 Properties of exponents
1.3.2 Simplifying expressions
1.3.3 Isolating variables and expressions
1.3.4 Greatest common factor (GCF) and least common multiple (LCM)
1.3.5 Standard form of linear equations
1.3.6 Key word operators
1.3.7 Arithmetic and geometric sequences
1.4 Intermediate algebra
1.5 Plane geometry
1.6 Coordinate geometry
1.7 Trigonometry
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3. ACT Reading
4. ACT Science
5. ACT Writing
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1.3.3 Isolating variables and expressions
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1. ACT Math
1.3. Elementary algebra

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 we can find what is equal to by solving for everything else in the equation, .

Sometimes, we are given equations that do not already have the key variable isolated. This could look like . 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 to the innermost parts in relation to , 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:

This is a simple equation with one variable. The purpose of isolating in this equation is to find the exact value of . Let’s start by figuring out how to get 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 that is added to the

  2. Get rid of the that is multiplied by

Which of these should we do first? Following reverse PEMDAS, or starting from the number furthest away from , we have to deal with the addition of 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

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

We have now successfully eliminated the added . 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 from the left side of the equation to get alone, you have to do the same to the right side.

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

Now simplifying the fractions, our final equation is

This is the purpose of isolating variables! We get to clearly identify the value of 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 in the equation ?

(spoiler)

First, add to both sides of the equation:

Now, divide by on both sides of the equation:

It is not an integer answer, but it is the value of !

What is the value of in the equation ?

(spoiler)

First, add to both sides of the equation:

Now, multiply by on both sides of the equation:

The value of is !

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

(spoiler)

Finding an expression means that we should isolate variables as we have been doing. There are two variables, so we need to pick the variable we want to isolate: . Now, we need to perform our operations to isolate . We will multiply by on both sides of the equation:

We can simplify this by distributing into the parentheses:

This is the expression for !

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 . As we increase the value of , the value of also increases. This is a function for .

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

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

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

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

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

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

Our final answer to this problem is .

Function notation

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

Evaluate for the following function:

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

Try another example.

Given the function , evaluate .

(spoiler)

Replace the variable () with the number used in the function ():

Simplify this equation to get the final answer:

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 . 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 . Let’s examine what this looks like:

Evaluate for the following functions:

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

Now, instead of using we will use . Let’s make that change:

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

This is our final answer! Just simplify to get the answer .

Let’s try another example.

Given:

Evaluate the function .

(spoiler)

Write out your work as the problem reads. First, :

Now, instead of using the problem uses . What is ? It is the function () using the value () as the variable:

Now that we know that is , we can put that into the function for :

So, our final answer to the problem is

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. is sometimes used instead of , and it effectively means “function using as the variable…”

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

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