The topic of linear equations is one of the most basic and important for any student who is beginning with the study of algebra. This article will help you to learn about linear equations or reinforce your previous knowledge. Specifically, we will look at linear equations with one variable along with several exercises to facilitate understanding the process used to solve the equations.

## Definition of linear equations

Equations are of linear when they can be written in the form , where is a variable and and are known constants. When a linear equation is reduced to its simplest form, it contains only variables raised to the first power.

Thus, the equations and are linear equations. The equation does not appear to be linear at first glance.

However, when we write it in its simplest form, combining like terms, the terms disappear and the equation reduces to . Therefore, this equation is linear.

There is exactly one solution to a linear equation with one variable.

### EXAMPLES

The following are linear equations:

•

•

•

•

## Rule of equations

We can perform any operation (addition, subtraction, multiplication, division) on one side of the equation, as long as we perform the same operation on the other side of the equation.

For example, suppose we have the equation .

If we wanted to, we could subtract 10 from both sides of the equation: .

Note that this time, it doesn’t help us much to have subtracted 10. When we solve equations, we want to perform operations in a way that simplifies the current equation and helps us solve the equation.

Now we are going to look at techniques for solving linear equations with one variable.

## Method for solving linear equations with one variable

1. Simplify each side:

a. Remove the parentheses (using the distributive property).

b. Eliminate the fractions (multiplying both sides by the least common multiple).

c. Combine like terms.

2. Solve for the variable:

This means adding/subtracting variables to get the variables to be on only one side of the equation and adding/subtracting numbers to get the numbers to be on the other side of the equation.

3. Perform operations so that the *x* is alone.

4. Check your answer.

Start now: Explore our additional Mathematics resources

## Linear equations with one variable – Examples with answers

### EXAMPLE 1

- Find the value of in the equation .

**Solution:**

1. We do not have parentheses or fractions. There are also no like terms to combine.

2. Isolate the variable: move the 5 to the right:

3. We perform operations so that the is alone: we divide both sides by 4:

4. Check your answer: we substitute the value in the original equation:

This is true

**Answer:**

### EXAMPLE 2

- Find the value of in the equation .

**Solution:**

1. There are no parentheses or fractions. We can combine like terms:

2. Solve for the variable: we move the 4 to the right and the to the left:

3. We perform operations so that the is alone: we divide both sides by 5:

4. Check your answer: we substitute the value in the original equation:

This is true

**Answer:**

### EXAMPLE 3

- Find the value of in the equation .

**Solution:**

1. Simplify:

We remove parentheses: .

There are no fractions.

We combine like terms: .

2. Solve for the variable: we move the 5 to the right:

3. We perform operations so that the is alone: we divide both sides by 7:

4. Check your answer: we replace the value in the original equation:

This is true

**Answer:**

### EXAMPLE 4

- Find the value of in the equation .

**Solution:**

1. Simplify:

We simplify the parentheses: .

We simplify fractions: .

We combine like terms: .

2. Solve for the variable: we move the 16 to the right and the -4x to the left:

3. We perform operations so that the is alone: we divide both sides by 13:

4. Check your answer: we replace the value in the original equation:

This is true

**Answer:**

## Linear equations with one variable – Practice problems

## See also

Interested in learning more about linear equations? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**