# How to Evaluate Expressions with Substitution

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will review the definition of an algebraic expression and order of operations. We will then look at how to evaluate algebraic expressions using substitution and practice the process on multiple examples.

## Finding Area

Suppose you are redecorating your house, and you need to know the area of a rectangular rug to know if it will be a good fit for your dining room. You know that to calculate the area of a rectangle, you need to multiply the length of the rectangle by the width. That is, area = l*w, where l = length and w = width. You use a tape measure to measure the rug's dimensions and find that the length is 11 feet and the width is 8 feet. To calculate the area of the rug, all we need to do is plug 11 for l and 8 for w into l*w, and simplify.

We see that the area of the rug is 88ft 2. This will fit perfectly in your dining room!

The process of plugging the length and width into the expression l*w is an example of evaluating an algebraic expression using substitution.

## Algebraic Expressions and PEMDAS

An algebraic expression is an expression containing more than one number, variable, and arithmetic operation. Some examples of algebraic expressions are shown.

We can evaluate an algebraic expression using substitution by simply plugging in values for the variables and simplifying in the appropriate order. That statement probably left you wondering what was meant by 'the appropriate order'. When we evaluate or simplify an algebraic expression, we have an order of operations that we follow. This order is illustrated in the following image:

A great way to remember this order is by remembering the word PEMDAS. As shown in the image, this word's letters each represent an operation, and they are in the correct order.

For example, consider the expression -3(1 + 4) - 82/4. To evaluate this expression, we don't just work left to right like we would read a book, we have to follow our order of operations. That is, we have to go in the order PEMDAS.

First, we address the P in PEMDAS, or the parentheses. Thus, the first thing we want to do is simplify what is in the parentheses to get -3(4 + 1) - 8 2/4 = -3(5) - 8 2/4. We also have a division bar, which is an inclusion, or grouping, symbol. This falls under P in PEMDAS as well, so we want to simplify the numerator and denominator to get -3(5) - 8 2/4 = -3(5) - 64/4.

At this point, there are no exponents, so we can move onto the M in PEMDAS. That is multiplication. We perform any multiplication we see to get -3(5) - 64/4 = -15 - 64/4.

The next letter in PEMDAS is D, which stands for division, so we perform any division in the expression to get -15 - 64/4 = -15 - 16.

There is no addition at this point, so we move onto the S in PEMDAS, which is subtraction. Thus, we have -15 - 16 = -31. This is all illustrated in a compact form in the image.

We see that if we evaluate the expression in the correct order, we get -31. Now that we've looked at the order in which we should simplify or evaluate expressions, let's look at evaluating algebraic expressions using substitution.

## Evaluating Algebraic Expressions Using Substitution

As we said, when we found the area of the rug for your dining room, we evaluated the algebraic expression l*w using substitution. We plugged values in for the variables and then evaluated. In general, to evaluate algebraic expressions using substitution, we follow these steps.

1. Plug in given values for variables in the expression.
2. Use PEMDAS to evaluate the expression.

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