Online College Courses for Credit

+
3 Tutorials that teach Solving single-step equations
Take your pick:
Solving single-step equations

Solving single-step equations

Author: Sophia Tutorial
Description:

Identify the operation needed to solve a single-step equation.

(more)
See More

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities.*

Begin Free Trial
No credit card required

46 Sophia partners guarantee credit transfer.

299 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 33 of Sophia’s online courses. Many different colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

Tutorial
what's covered
  1. Solving single-step equations
    1. Addition Problems
    2. Subtraction Problems
    3. Multiplication Problems
    4. Division Problems

1. Solving single-step equations

Solving linear equations is an important and fundamental skill in algebra. In algebra, we are often presented with a problem where the answer is known, but part of the problem is missing. The missing part of the problem is what we seek to find. An example of such a problem is shown below.

4 x plus 16 equals short dash 4

Notice the above problem has a missing part, or unknown, that is marked by x. If we are given that the solution to this equation is −5, it could be plugged into the equation, replacing the x with −5. This is shown in below:

4 open parentheses short dash 5 close parentheses plus 16 equals short dash 4
Multiply 4 open parentheses short dash 5 close parentheses
short dash 20 plus 16 equals short dash 4
Add short dash 20 plus 16
short dash 4 equals short dash 4
True!

Now the equation comes out to a true statement! Notice also that if another number, for example, 3, was plugged in, we would not get a true statement.

4 open parentheses 3 close parentheses plus 16 equals short dash 4
Multiply 4 open parentheses 3 close parentheses
12 plus 16 equals short dash 4
Add 12 plus 16
28 not equal to short dash 4
False!

Due to the fact that this is not a true statement, this demonstrates that 3 is not the solution. However, depending on the complexity of the problem, this “guess and check” method is not very efficient. Thus, we take a more algebraic approach to solving equations. Here we will focus on what are called “one-step equations” or equations that only require one step to solve. While these equations often seem very fundamental, it is important to master the pattern for solving these problems so we can solve more complex problems.

1a. Addition Problems

To solve equations, the general rule is to do the opposite. For example, consider the following example.

x plus 7 equals short dash 5
The 7 is added to the x
space space space space space space space minus 7 space space space space space space minus 7
Subtract 7 from both sides to get rid of it
x equals short dash 12
Our Solution

Then we get our solution, x = − 12. The same process is used in each of the following examples.

4 plus x equals 8
7 equals x plus 9
5 equals 8 plus z
stack negative 4 space space space space minus 4 with bar below
stack negative 9 space space space space minus 9 with bar below
stack negative 8 space space space minus 8 with bar below
x equals 4
short dash 2 equals x
short dash 3 equals z

1b. Subtraction Problems

In a subtraction problem, we get rid of negative numbers by adding them to both sides of the equation. For example, consider the following example.

x minus 5 equals 4
The 5 is negative, or subtracted from z
stack space space plus 5 space space plus 5 with bar below
Add 5 to both sides
x equals 9
Our Solution

Then we get our solution x = 9. The same process is used in each of the following examples. Notice that each time we are getting rid of a negative number by adding.

short dash 6 plus x equals short dash 2
short dash 10 equals x minus 7
5 equals short dash 8 plus z
stack plus 6 space space space space space space space space space space plus 6 with bar below
stack plus 7 space space space space plus 7 with bar below
stack plus 8 space space space plus 8 with bar below
x equals 4
short dash 3 equals x
13 equals x

1c. Multiplication Problems

With a multiplication problem, we get rid of the number by dividing on both sides. For example consider the following example.

4 x equals 20
Variable is multiplied by 4
stack space 4 space with bar on top space space space space stack space 4 space with bar on top
Divide both sides by 4
x equals 5
Our Solution

Then we get our solution x=5

With multiplication problems it is very important that care is taken with signs. If x is multiplied by a negative then we will divide by a negative. This is shown in the next example:

short dash 5 x equals 30
Variable is multiplied by short dash 5
stack space short dash 5 with bar on top space space space space stack short dash 5 space with bar on top
Divide both sides by short dash 5
x equals short dash 6
Our Solution

The same process is used in each of the following examples. Notice how negative and positive numbers are handled as each problem is solved.

8 x equals short dash 24
short dash 4 x equals short dash 20
42 equals 7 x
stack space 8 space with bar on top space space space space space space space space stack space 8 space with bar on top
stack space short dash 4 space with bar on top space space space space space space stack space short dash 4 space with bar on top
stack space 7 space with bar on top space space space space space stack space 7 space with bar on top
x equals short dash 3
x equals 5
6 equals x

1d. Division Problems

In division problems, we get rid of the denominator by multiplying on both sides. For example consider our next example.

x over 5 equals short dash 3
Variable is divided by 5
open parentheses 5 close parentheses x over 5 equals short dash 3 open parentheses 5 close parentheses
Multiply both sides by 5
x equals short dash 15
Our Solution

Then we get our solution x = − 15. The same process is used in each of the following examples.

fraction numerator x over denominator short dash 7 end fraction equals short dash 2
x over 8 equals 5
fraction numerator x over denominator short dash 4 end fraction equals 9
open parentheses short dash 7 close parentheses fraction numerator x over denominator short dash 7 end fraction equals 2 open parentheses short dash 7 close parentheses
open parentheses 8 close parentheses x over 8 equals 5 open parentheses 8 close parentheses
open parentheses short dash 4 close parentheses fraction numerator x over denominator short dash 4 end fraction equals 9 open parentheses short dash 4 close parentheses
x equals 14
x equals 40
x equals short dash 36

The process described above is fundamental to solving equations. Once this process is mastered, the problems we will see have several more steps. These problems may seem more complex, but the process and patterns used will remain the same.

big idea
To solve single-step equations, first identify the operation being applied to the variable. To isolate the variable, apply the inverse operation. Addition and subtraction are inverses of each other; multiplication and division are inverses of each other.

summary
Solving single-step equations involves isolating the variable that you're trying to solve for by using an inverse operation. Any operation that you do on one side of the equation needs to be done on the other side. You can always check your solution by substituting it in for the variable in your original equation and seeing if the statement holds true. This is good practice to get into when you're doing simple equations because when you do things more complicated, it's more likely that you're going to be making a mistake.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html

Terms to Know
Equation

a mathematical statement that two quantities or expressions are equal in value

Rule of Equality

any operation performed on one side of the equation must be performed on the other side, in order to keep quantities equal in value