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:
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.
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.
Now simplify addition and subtraction in the order they appear (combine like terms):
Substitution
Simplifying Before Substituting
As you can see, the values of b and c are different for this problem:
b^{2}c + bc^{2} + 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 “2b^{2}c”.
Now combine like terms: b^{2}c and 2b^{2}c are combined into 3b^{2}c.
The problem is now simplified to the following:
3b^{2}c + bc^{2}
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).
Use the Order of Operations to simplify the expression. First simplify exponents.
Simplify multiplication.
54 – 12
Combine like terms.
Substitution
If the value for only one of the variables is given
Look over the problem below.
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.
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.
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:
Simplify:
Substitution Introduction Resources
Practice Problems / Worksheet Order of Operations Calculator

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