3.1: OneStep Equations and Inverse Operations
What if you were in a math contest and were given the equation
Guidance
It’s Easier than You Think
You have been solving equations since the beginning of this textbook, although you may not have recognized it. For example, in a previous Concept, you determined the answer to the pizza problem below.
$20.00 was onequarter of the money spent on pizza.
The solution is 80. So, the amount of money spent on pizza was $80.00.
By working through this question mentally, you were applying mathematical rules and solving for the variable
Definition: To solve an equation means to write an equivalent equation that has the variable by itself on one side. This is also known as isolating the variable.
In order to begin solving equations, you must understand three basic concepts of algebra: inverse operations, equivalent equations, and the Addition Property of Equality.
Inverse Operations and Equivalent Equations
In another Concept, you learned how to simplify an expression using the Order of Operations: Parentheses, Exponents, Multiplication and Division completed in order from left to right, and Addition and Subtraction (also completed from left to right). Each of these operations has an inverse. Inverse operations “undo” each other when combined.
For example, the inverse of addition is subtraction. The inverse of an exponent is a root.
Definition: Equivalent equations are two or more equations having the same solution.
The Addition Property of Equality
Just like Spanish, chemistry, or even music, mathematics has a set of rules you must follow in order to be successful. These rules are called properties, theorems, or axioms. They have been proven or agreed upon years ago, so you can apply them to many different situations.
For example, the Addition Property of Equality allows you to apply the same operation to each side of an equation, or “what you do to one side of an equation you can do to the other.”
The Addition Property of Equality
For all real numbers
If
Solving OneStep Equations Using Addition or Subtraction
Because subtraction can be considered “adding a negative,” the Addition Property of Equality also works if you need to subtract the same value from each side of an equation.
Example A
Solve for
Solution: When asked to solve for
Write the original equation:
Apply the Addition Property of Equality:
Simplify by adding like terms:
The solution is
Example B
Solve for
Solution:
Apply the Addition Property of Equality:
The solution is
Equations that take one step to isolate the variable are called onestep equations. Such equations can also involve multiplication or division.
Solving OneStep Equations Using Multiplication or Division
The Multiplication Property of Equality
For all real numbers
If
Example C
Solve for
Solution: Because
Write the original equation:
Apply the Multiplication Property of Equality:
The solution is
When working with fractions, you must remember:
Vocabulary
Solving equations: To solve an equation means to write an equivalent equation that has the variable by itself on one side. This is also known as isolating the variable.
Inverse operation: Each of these operations has an inverse. Inverse operations “undo” each other when combined.
Equivalent equation: By applying the same inverse operations to each side of an equation, you create an equivalent equation. Equivalent equations are two or more equations having the same solution.
Onestep equations: Equations that take one step to isolate the variable are called onestep equations. Such equations can also involve multiplication or division.
The Addition Property of Equality
For all real numbers
If
The Multiplication Property of Equality
For all real numbers
If
Guided Practice
1. Determine the inverse of division.
2. Solve
Solutions:
1. To undo division by a number, you would multiply by the same number.
2. The variable
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: OneStep Equations (12:30)
Solve for the given variable.

x+11=7 
x−1.1=3.2 
7x=21 
4x=1 
5x12=23 
x+52=23 
x−56=38 
0.01x=11  \begin{align*}q  13 = 13\end{align*}
 \begin{align*}z + 1.1 = 3.0001\end{align*}
 \begin{align*}21s = 3\end{align*}
 \begin{align*}t + \frac{1}{2} = \frac{1}{3}\end{align*}
 \begin{align*}\frac{7f}{11} = \frac{7}{11}\end{align*}
 \begin{align*}\frac{3}{4} =  \frac{1}{2} \cdot y\end{align*}
 \begin{align*}6r = \frac{3}{8}\end{align*}
 \begin{align*}\frac{9b}{16} = \frac{3}{8}\end{align*}
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Addition Property of Equality
For all real numbers and : If , then .equivalent equation
By applying the same inverse operations to each side of an equation, you create an equivalent equation. Equivalent equations are two or more equations having the same solution.inverse operation
Each of these operations has an inverse. Inverse operations undo each other when combined.Multiplication Property of Equality
For all real numbers , and : If , thenonestep equations
Equations that take one step to isolate the variable. Such equations can also involve multiplication or division.solving equations
To solve an equation means to write an equivalent equation that has the variable by itself on one side. This is also known as isolating the variable.constant
A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.Numerical Coefficient
In mathematical expressions, the numerical coefficients are the numbers associated with the variables. For example, in the expression , 4 is the numerical coefficient and is the variable.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Image Attributions
Here you'll learn how to isolate variables using inverse operations in order to solve equations in one step.