# 2.2.1: One-Step Equations and Properties of Equality

- Page ID
- 4352

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## One-Step Equations and Inverse Operations

What if you were in a math contest and were given the **equation** x+4=16? Or how about the equation 9x=72? Could you solve each of these equations in one step?

**One-Step Equations** and Inverse Operations

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.**

**Solving One-Step Equations Using Addition or Subtraction**

There are four important ideas and properties that you need to **solving equations** using addition or subtraction. The order of operations, inverse operations, equivalent equations, and the **addition property of equality**.

You probably have already learned how to simplify an expression using the Order of Operations: **P**arentheses, **E**xponents, **M**ultiplication and **D**ivision completed in order from left to right, and **A**ddition and **S**ubtraction (also completed from left to right).

Each of these operations has an **inverse. **Inverse operations “undo” each other when combined and are essential to solving equations. Addition and subtraction are inverses, multiplication and division are inverses, and exponents and roots are inverses.

**Equivalent equations** are two or more equations with the same solution.

The last tool you need to solve equations using addition or subtraction is the Addition Property of Equality. The **Addition Property of Equality** allows you to apply the same operation to each side of an equation, in essence: “what you do to one side of an equation you can do to the other.”

The Addition Property of Equality states that for all real numbers a,b, and c:

If a=b, then a+c=b+c.

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.

#### Let's solve the following equations using the Addition Property of Equality:

- Solve for y:

16=y−11.

When asked to solve for y, your goal is to write an equivalent equation with the variable y isolated on one side.

Write the original equation: 16=y−11.

The inverse of subtracting 11 is adding 11.

Apply the Addition Property of Equality: 16+11=y−11+11.

Simplify by adding like terms: 27=y.

The solution is y=27.

- Solve for z:

5=z+12

The inverse of adding 12 is subtracting 12.

Apply the Addition Property of Equality:

5=z+12

5−12=z+12−12

5−12=z

−7=z

The solution is −7=z.

Equations that take one step to isolate the variable are called **one-step equations**. Such equations can also involve multiplication or division.

**Solving One-Step Equations Using Multiplication or Division**

When solving one-step equations that involve multiplication or division, you will be using inverses and equivalent equations. However, instead of using the Addition Property of Equality, you will use the very similar **Multiplication Property of Equality**. The **Multiplication Property of Equality **states that for all real numbers a,b, and c:

If a=b, then a(c)=b(c).

Since division thought of as multiplying by the reciprocal, the Multiplication Property of Equality also applies to dividing both sides of the equation by the same number.

**Let's solve the following equation using the Multiplication Property of Equality:**

Solve for k:−8k=−96.

Because −8k=−8×k, the **inverse operation** of multiplication is division. Therefore, we must cancel multiplication by applying the Multiplication Property of Equality.

Write the original equation: −8k=−96.

The inverse of multiplying by 8 is dividing by 8.

Apply the Multiplication Property of Equality: −8k÷−8=−96÷−8.

The solution is k=12.

When working with fractions, you must remember: a/b×b/a=1. In other words, in order to cancel a fraction using division, you really must multiply by its reciprocal.

**Examples**

Example 2.2.1.1

Earlier, you were asked to solve the equations x+4=16 and 9x=72.

x+4=16

**Solution**

ince this is an addition problem, we will be applying the Addition Property of Equality. The inverse of adding 4 is subtracting 4.

x+4=16

x+4−4=16−4

x=12

The solution for x+4=16 is x=12.

9x=72

Since this is a multiplication problem, we will be applying the Addition Property of Equality. The inverse of multiplying by 9 is dividing by 9.

9x=72

9x÷9=72÷9

x=8

The solution for 9x=72 is x=8.

Example 2.2.1.2

Solve 18⋅x=1.5.

**Solution**

The variable x is being multiplied by one-eighth. Instead of dividing two fractions, we multiply by the reciprocal of 1/8, which is 8.

8(1/8⋅x=8(1.5)

x=12

### Review

Solve for the given variable.

- x+11=7
- x−1.1=3.2
- 7x=21
- 4x=1
- 5x/12=2/3
- x+5/2=2/3
- x−5/6=3/8
- 0.01x=11
- q−13=−13
- z+1.1=3.0001
- 21s=3
- t+1/2=1/3
- 7f/11=7/11
- 3/4=−1/2⋅y
- 6r=3/8
- 9b/16=3/8

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.1.

### Vocabulary

Term | Definition |
---|---|

Addition Property of Equality |
For all real numbers a,b, and c: If a=b, then a+c=b+c. |

equivalent equation |
By applying the same inverse operations to each side of an equation, you create an . equivalent equation are two or more equations having the same solution.Equivalent equations |

inverse operation |
Each of these operations has an Inverse operations inverse. each other when combined.undo |

Multiplication Property of Equality |
For all real numbers a,b, and c: a=b, If a(c)=b(c).then |

one-step equations |
Equations that take one step to isolate the variable. Such equations can also involve multiplication or division. |

solving equations |
To an equation means to write an equivalent equation that has the variable by itself on one side. This is also known as solveisolating 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 or x, y.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 4x, 4 is the numerical coefficient and x 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. |

### Additional Resources

Video:

PLIX: Play, Learn, Interact, eXplore: **One-Step Equations Transformed by MultiplicationDivision: Dividing **(...)

Activity: **One-Step Equations and Inverse Operations Discussion Questions**

Practice: **One-Step Equations and Properties of Equality**

Real World Application: **The Cost of College**