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Substitution Lessons

In algebra, letters such as x or y are used to represent values which are usually unknown. They can be used in equations or expressions to help solve a wide variety of problems. In many cases you may know the value of a variable. This is the case with the problem below:

b = 3, c = 18
5b - 2c + c / b

Since the values of both b and c are known, a numeric value for the expression can be found through the process of substitution.

When a school teacher becomes ill, a substitute teacher usually fills in for a few days. The substitute teacher temporarily takes the place of the usual teacher so that classroom activities may continue. Similarly, when using the method of substitution in algebra, a variable such as x or y is replaced with its value. The expression can then be simplified even further.

In this problem we replace the variables b and c since their values are given. Everywhere in the problem where the variable b is present, it is substituted with 3 in parentheses. Everywhere c appears, it is substituted with 18 in parentheses. It is important that the substituted number is placed in parentheses so that negative values are handled properly.

5(3) - 2(18) + (18) / (3)

Now, this expression can be simplified like any other.

The insides of all of the parentheses are simplified, and there are no exponents in the expression, so you can skip to simplifying multiplication and division in the order they appear.

15 - 36 + 6

Now simplify addition and subtraction in the order they appear (combine like terms):

-15

Substitution

Simplifying Before Substituting

As you can see, the values of b and c are different for this problem:

b = -3, c = 2
b2c + bc2 + 2b · bc

In the previous example, the expression could not be simplified before its variables, b and c, were substituted with their values.

Whenever possible, an expression should be simplified before substitution is applied, as it will often save time. Begin by simplifying multiplication: "2b · bc" becomes "2b2c".

b2c + bc2 + 2b2c

Now combine like terms: b2c and 2b2c are combined into 3b2c.

3b2c + bc2

The problem is now simplified to the following:

b = -3, c = 2
3b2c + bc2

We will begin the substition in a moment, but first compare this expression to the expression we started with. Both expressions are equivalent because we simplified properly. But, by reducing the number of terms in the expression, the substitution step will be easier.

Now substitute b with (-3) and c with (2).

3(-3)2(2) + (-3)(2)2

Use the Order of Operations to simplify the expression. First simplify exponents.

3(9)(2) + (-3)(4)

Simplify multiplication.

3(18) + -12
54 - 12

Combine like terms.

42

Substitution

If the value for only one of the variables is given

Look over the problem below.

b = 3
5b - 2c + c / b

As you can see, the expression has two variables b and c, but in this problem, only the value for b is given. Don't panic! Simply substitute the variable b with the number 3, don't worry about doing anything with the variable c.

5(3) - 2c + c / (3)

Then simplify as much as you can using the Order of Operations:


If the value for one of the variables is not a number

In this case, as in the problem below, you must substitute the variable with the expression given.

b = c + 1
5b - 2c + c / b

NOTE

This problem is one instance where it is absolutely necessary to enclose the expression or number that you are substituting a variable for in parentheses. Because of the Order of Operations 5(c + 1) is not the same as 5c + 1.

Substitute:

5(c + 1) - 2c + c / (c + 1)

Simplify:

5c + 5 - 2c + c / (c + 1)

Substitution Introduction Resources

Practice Problems / Worksheet
The worksheet for this lesson.

Order of Operations Calculator
After you have substituted an expression's variables, use this calculator to simplify the expression for you.

Next Lesson: Graphing Linear Equations
Graphs provide a visual representation of the relationship between two variables. In this lesson, learn how to graph and solve two variable equations, and become comfortable with coordinate planes, ordered pairs, and more.

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