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One-Step Equations

Introduction

An equation is a mathematical statement that demonstrates the equality of two expressions. You will frequently work with equations in algebra where a variable represents an unknown value. To solve these equations, it would be best to determine the variable’s value. To find the unknown value in a one-step equation, you only need to perform one operation; hence, this problem is the simplest to solve.

This article will teach us about one-step equations and explain how to solve cases using various examples.

What is a One-Step Equation?

Definition

An algebraic equation that only requires one step to solve is known as a one-step equation. Once the equation has been solved, we can find the value of the variable that makes the statement true.

We can use addition, subtraction, multiplication, or division to solve a one-step equation.

Types of One-Step Equations

The following are the different types of one-step equations:

  1. One-step equations with subtraction

Examples: 

x – 4 = 7y – 6 = 10
k – 5 = 12m – 20 = 31
  1. One-step equations with addition

Examples: 

k + 1 = 13a + 9 = 18
r + 10 = 16c + 12 = 25
  1. One-step equations with division

Examples: 

$\frac{x}{3}$=4$\frac{b}{2}$=9
$\frac{a}{4}$=5$\frac{s}{5}$=10
  1. One-step equations with multiplication

Examples: 

6k = 123y = 30
4x = 2410p = 100

How do you solve one-step equations?

When solving one-step equations, we conduct the opposite operation from that applied to the variable to obtain the variable by itself.

These are the inverse operations:

Addition and Subtraction
Multiplication and Division

Let us say, for example, use subtraction when there is addition, use addition when there is subtraction, use multiplication when there is division, and vice versa.

Important: Any changes you make to one side of the equation must also be made to the other side. You have just solved the equation if you can isolate the variable by itself so that the variable has a +1 coefficient, and the constant is on the opposite side.

Solving One-Step Equations with Addition

To solve a one-step equation with addition of a number, we must subtract this number from both sides of the equation.

Examples

Example 1

Solve the one-step equation: x + 5 = 18.

Solution

In this example, x has 5 added to it. Since we want to isolate the variable x on the left-hand side of the equation, we will perform the inverse operation and subtract 5 from both sides of the equation. Hence, we have,

x + 5 = 18 Given
x + 5 – 5 = 18 – 5 Subtract 5 from both sides of the equation
x = 13 Simplify

Checking

To check the answer, substitute 13 into the variable x and see if the equation is true:

 x + 5 = 18
13 + 5 = 18
18 = 18

Since the equation is true, x = 13 is the correct solution.

Example 2

Solve for y in the following equation: y + 15 = 45.

Solution

In this example, y has 15 added to it. Since we want to isolate the variable y on the left-hand side of the equation, we will perform the inverse operation and subtract 15 from both sides of the equation. Thus, we have,

y + 15 = 45 Given
y + 15 – 15 = 45 – 15 Subtract 15 from both sides of the equation
y = 30 Simplify

Checking

To check the answer, substitute 30 into the variable y and see if the equation is true:

 y + 15 = 45
30 + 15 = 45
45 = 45

Since the equation is true, y = 30 is the correct solution.

Example 3

Solve for k: 

k + 12 = 28

Solution

In this example, k has 12 added to it. Since we want to isolate the variable k on the left-hand side of the equation, we will perform the inverse operation and subtract 12 from both sides of the equation. Hence, we have,

k + 12 = 28 Given
k + 12 – 12 = 28 – 12 Subtract 12 from both sides of the equation
k = 16 Simplify

Checking

To check the answer, substitute 16 into the variable k and see if the equation is true:

 k + 12 = 28
16 + 12 = 28
28 = 28

Since the equation is true, k = 16 is the correct solution.

Example 4

Solve the one-step equation below.

36 = k + 7

Solution

Notice that this example is slightly different from the previous examples since the variable is on the right-hand side of the equation. Remember that we need to isolate the variable on either side of the equation. Since k has 7 added to it, we will perform the inverse operation and subtract 8 from both sides of the equation. Hence, we have,

Given

36 = k + 7
Seven must be subtracted from both sides of the equation
36 – 7 = k + 7 – 7

Simplify

29 = k

Checking

To check the answer, substitute 29 into the variable k and see if the equation is true:

36 = k + 
7
36 = 29 + 7 
36 = 36

Since the equation is true, k = 29 is the correct solution.

Solving One-Step Equations with Subtraction

To solve a one-step equation with subtraction of a number, we must subtract this number from both sides of the equation.

Examples

Example 1

Solve the one-step equation: y – 6 = 25.

Solution

Since we want to isolate the variable y on the left-hand side of the equation, we will perform the inverse operation and add 6 to both sides of the equation. Hence, we have,

y – 6 = 25 Given
y – 6 + 6  = 25 + 6 Add 6 to both sides of the equation
y = 31 Simplify

Checking

To check the answer, substitute 31 into the variable y and see if the equation is true:

y – 6 = 25
31 – 6 = 25
25 = 25

Since the equation is true, y = 31 is the correct solution.

Example 2

Solve for m in the following equation: m – 21 = 35.

Solution

Since we want to isolate the variable m on the left-hand side of the equation, we will perform the inverse operation and add 21 to both sides of the equation. Thus, we have,

m – 21 = 35 Given
m – 21 + 21  = 35 + 21 Add 21 to both sides of the equation
m = 56 Simplify

Checking

To check the answer, substitute 56 into the variable y and see if the equation is true:

m – 21 = 35
56 – 21 = 35
35 = 35

Since the equation is true, m = 56 is the correct solution.

Example 3

Solve for x: 

x – 17 = 53

Solution

Since we want to isolate the variable x on the left-hand side of the equation, we will perform the inverse operation and add 17 to both sides of the equation. Hence, we have,

x – 17 = 53 Given
x – 17 + 17  = 53 + 17 Add 17 to both sides of the equation
x = 70 Simplify

Checking

To check the answer, substitute 70 into the variable x and see if the equation is true:

x – 17 = 53
70 – 17 = 53
53 = 53

Since the equation is true, x = 70 is the correct solution.

Example 4

Solve the one-step equation below.

112 = k – 48 

Solution

Notice that this example is slightly different from the previous examples since the variable is on the right-hand side of the equation. Remember that we need to isolate the variable on either side of the equation. We will perform the inverse operation and add 48 from both sides of the equation. Hence, we have,

112 = k – 48  Given
112 + 48 = k – 48 + 48 Add 48 from both sides of the equation
160 = k Simplify

Checking

To check the answer, substitute 160 into the variable k and see if the equation is true:

112 = k – 48 
112 = 160 – 48  
112 = 112

Since the equation is true, k = 160 is the correct solution.

Solving One-Step Equations with Multiplication

A one-step equation involving multiplication has a number ( other than 1 ) in front of the variable. To solve it, we must multiply both sides of this equation by this number.

Examples

Example 1

Solve the one-step equation: 5b = 45.

Solution

In this example, 5b implies 5 times b, which involves multiplication. Since we want to isolate the variable b on the left-hand side of the equation, we will perform the inverse operation and divide both sides of the equation by 5. Hence, we have,

5b = 45 Given

$\frac{5b}{5}$  = $\frac{45}{5}$  Divide both sides of the equation by 5

b = 9 Simplify

Checking

To check the answer, substitute 9 into the variable b and see if the equation is true:

5b = 45
5 × 9 = 45
45 = 45

Since the equation is true, b = 9 is the correct solution.

Example 2

Solve for y in the following equation: 8y = 96.

Solution

In this example, 8y implies 8 times y, which involves multiplication. Since we want to isolate the variable y on the left-hand side of the equation, we will perform the inverse operation and divide both sides of the equation by 8. Thus, we have,

8y = 96 Given

$\frac{8y}{8}$  = $\frac{96}{8}$  Divide both sides of the equation by 8

y = 12 Simplify

Checking

To check the answer, substitute 12 into the variable y and see if the equation is true:

8y = 96
8 × 12 = 96
96 = 96

Since the equation is true, y = 12 is the correct solution.

Example 3

Solve for c: 

7c = 63

Solution

In this example, 7c implies 7 times c, which involves multiplication. Since we want to isolate the variable c on the left-hand side of the equation, we will perform the inverse operation and divide both sides of the equation by 7. Thus, we have,

7c = 63 Given

$\frac{7c}{7}$  = $\frac{63}{7}$  Divide both sides of the equation by 7

c = 9 Simplify

Checking

To check the answer, substitute 9 into the variable c and see if the equation is true:

7c = 63
7 × 9 = 63
63 = 63

Since the equation is true, c = 9 is the correct solution.

Example 4

Solve the one-step equation below.

168 = 12d

Solution

In this example, 12d implies 12 times d, which involves multiplication. Notice that it is slightly different from the previous examples since the variable is on the right-hand side of the equation. Remember that we need to isolate the variable on either side of the equation. We will perform the inverse operation and divide both sides by 12. Thus, we have,

168 = 12d Given

$\frac{168}{12}$ =$\frac{12d}{12}$  Divide both sides of the equation by 12

14 = d Simplify

Checking

To check the answer, substitute 14 into the variable d and see if the equation is true:

168 = 12d 
168 = 12 × 14  
168 = 168

Since the equation is true, d = 14 is the correct solution.

Solving One-Step Equations with Division

A one-step equation with division involves a fraction in which the numerator is a variable, and a denominator is a number. To solve a one-step equation with division, we must multiply both sides by the number in the fraction’s denominator.

Examples

Example 1

Solve the one-step equation: $\frac{k}{8}$ = 16.

Solution

In this example, $\frac{k}{8}$ implies k over 8, which involves division. Since we want to isolate the variable k on the left-hand side of the equation, we will perform the inverse operation and multiply both sides of the equation by 8. Hence, we have,

$\frac{k}{8}$ = 16 Given

$\frac{k}{8}$ × 8  = 16 × 8 Multiply each side of the equation by 8

k = 128 Simplify

Checking

To check the answer, substitute 128 into the variable k and see if the equation is true:

$\frac{k}{8}$ = 16

$\frac{128}{8}$ = 16

16 = 16

Since the equation is true, k = 128 is the correct solution.

Example 2

Solve for y in the following equation: $\frac{n}{11}$ = 13.

Solution

In this example, $\frac{n}{11}$ implies n over 13, which involves division. Since we want to isolate the variable n on the left-hand side of the equation, we will perform the inverse operation and multiply both sides of the equation by 11. Hence, we have,

$\frac{n}{11}$ = 13 Given

$\frac{n}{11}$ × 11  = 13 × 11 Multiply both sides of the equation by 11

n = 143 Simplify

Checking

To check the answer, substitute 143 into the variable n and see if the equation is true:

$\frac{n}{11}$ = 13

$\frac{143}{11}$ = 13

13 = 13

Since the equation is true, n = 143 is the correct solution.

Example 3

Solve for c: 

$\frac{c}{6}$=7

Solution

In this example, $\frac{6}{6}$ implies c over 6, which involves division. Since we want to isolate the variable c on the left-hand side of the equation, we will perform the inverse operation and multiply both sides of the equation by 6. Hence, we have,

$\frac{c}{6}$ = 7 Given

$\frac{c}{6}$ × 6  = 7 × 6 Multiply both sides of the equation by 6

c = 42 Simplify

Checking

To check the answer, substitute 42 into the variable c and see if the equation is true:

$\frac{c}{6}$ = 7

$\frac{42}{6}$ = 7

7 = 7

Since the equation is true, c = 42 is the correct solution.

Example 4

Solve the one-step equation below.

19=$\frac{w}{5}$

Solution

In this example, $\frac{w}{5}$ implies w over 5, which involves division. Notice that it is slightly different from the previous examples since the variable is on the right-hand side of the equation. Remember that we need to isolate the variable on either side of the equation. We will perform the inverse operation and multiply both sides by 5. Thus, we have,

19=$\frac{w}{5}$ Given

19 ( 5 )= $\frac{w}{5}$ ( 5 )  Multiply each side by 5

95 = w Simplify

Checking

To check the answer, substitute 95 into the variable w and see if the equation is true:

19=$\frac{w}{5}$

19=$\frac{95}{5}$

19 = 19

Since the equation is true, w = 95 is the correct solution.

Summary

An algebraic equation that only requires one step to solve is known as a one-step equation. Once the equation has been solved, we can find the value of the variable that makes the statement true.

We can use addition, subtraction, multiplication, or division to solve a one-step equation.

Types of one-step equations

  1. One-step equations with addition
  2. One-step equations with subtraction
  3. One-step equations with multiplication
  4. One-step equations with division

When solving one-step equations, we conduct the opposite operation from that being applied to the variable to obtain the variable by itself.

These are the inverse operations:
Addition and Subtraction
Multiplication and Division

Let us say, for example, use subtraction when there is addition, use addition when there is subtraction, use multiplication when there is division, and vice versa.

Important: Any changes you make to one side of the equation must also be made to the other side.

Frequently Asked Questions on One-Step Equations ( FAQs )

What are one-step equations?

Algebraic equations that only require one step to solve are known as one-step equations. We can use addition, subtraction, multiplication, or division to solve a one-step equation. When solving one-step equations, we conduct the opposite operation from that being applied to the variable to obtain the variable by itself.

What is the formula for solving one-step equations?

When solving one-step equations, we conduct the opposite operation from that being applied to the variable to obtain the variable by itself.

These are the inverse operations:
Addition and Subtraction
Multiplication and Division

Thus, use subtraction when there is addition, use addition when there is subtraction, use multiplication when there is division, and vice versa.

How do you solve one-step equations with a fraction as the coefficient?

To solve a one-step equation with a fraction as the coefficient, we must multiply each side of the one-step equation by the reciprocal of the fraction.

Let us say, for example, $\frac{4}{5}$k=20.

To find the solution, each side of the equation must be multiplied by the reciprocal of $\frac{4}{5}$, which is $\frac{5}{4}$. It will result in having the coefficient of k equal to 1.

Here is the math,

$\frac{4}{5}$k=20 Given

($\frac{4}{5}$k)($\frac{5}{4}$)=20 ($\frac{5}{4}$) Multiply both sides by $\frac{5}{4}$

k=25 Simplify

Checking

To check the answer, substitute 25 into the variable k and see if the equation is true:

 $\frac{4}{5}$k=20

$\frac{4}{5}$(25)=20

20 = 20

Since the equation is true, k = 20 is the correct solution.

How do you solve one-step equations with addition?

To solve a one-step equation with addition of a number, we must subtract this number from both sides of the equation.

y + 15 = 45 Given
y + 15 – 15 = 45 – 15 Subtract 15 from both sides of the equation
y = 30 Simplify

How do you solve one-step equations with subtraction?

To solve a one-step equation with subtraction of a number, we must subtract this number from both sides of the equation.

y – 6 = 25 Given
y – 6 + 6  = 25 + 6 Add 6 to both sides of the equation
y = 31 Simplify

Checking

To check the answer, substitute 31 into the variable y and see if the equation is true:

y – 6 = 25
31 – 6 = 25
25 = 25

Since the equation is true, y = 31 is the correct solution.

How do you solve one-step equations with multiplication?

A one-step equation involving multiplication has a number ( except 1 ) in front of the variable. To solve it, we must multiply both sides of this equation by this number.

5b = 45 Given

$\frac{5b}{5}$  = $\frac{45}{5}$  Divide both sides of the equation by 5

b = 9 Simplify

How do you solve one-step equations with division?

A one-step equation with division involves a fraction in which the numerator is a variable, and a denominator is a number. To solve a one-step equation with division, we must multiply both sides by the number in the fraction’s denominator.

$\frac{n}{11}$ = 13 Given
$\frac{n}{11}$ × 11  = 13 × 11 Multiply both sides of the equation by 11
n = 143 Simplify

Checking

To check the answer, substitute 143 into the variable n and see if the equation is true:

$\frac{n}{11}$ = 13
$\frac{143}{11}$ = 13
13 = 13
Since the equation is true, n = 143 is the correct solution.

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