Linear equations with one variable are equations with a single unknown, which is not raised to any power. These equations are solved by applying various operations to both sides of the equation to isolate the variable completely.

In this article, we will look at how to solve the linear equations with one variable along with several examples to facilitate understanding.

## 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 $latex 2x+5=5x-20$.

If we wanted to, we could subtract 10 from both sides of the equation: $latex 2x+5-10=5x-20-10$.

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

Equations are linear when they can be written in the form $latex ax+b=c$, where $latex x$ is a variable and $latex a, b$, and $latex c$ are known constants. When a linear equation is reduced to its simplest form, it contains only variables raised to the first power.

There is exactly one solution to a linear equation with one unknown. To solve this type of equation, we can follow the following steps:

**Step 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.

**Step 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.

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

**Step 4.** Check your answer.

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## Linear equations with one variable – Examples with answers

### EXAMPLE 1

Find the value of $latex x$ in the equation $latex 4x+5=17$.

##### 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:

$latex 4x+5-5=17-5$

$latex 4x=12$

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

$latex \frac{4}{4}x=\frac{{12}}{4}$

$latex x=\frac{{12}}{4}=3$

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

$latex 4\left( 3 \right)+5=17$

$latex 12+5=17$

$latex 17=17$

This is true

**Answer:** $latex x=3$.

### EXAMPLE 2

Find the value of $latex x$ in the equation $latex 2x+6-2=-3x+14$.

##### Solution

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

$latex 2x+4=-3x+14$

2. Solve for the variable: we move the 4 to the right and the $latex -3x$ to the left:

$latex 2x+4-4=-3x+14-4$

$latex 2x=-3x+10$

$latex 2x+3x=-3x+10+3x$

$latex 5x=10$

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

$latex \frac{5}{5}x=\frac{{10}}{5}$

$latex x=\frac{{10}}{5}=2$

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

$latex 2(2)+6-2=-3(2)+14$

$latex 4+6-2=-6+14$

$latex 8=8$

This is true

**Answer:** $latex x=2$.

### EXAMPLE 3

Find the value of $latex x$ in the equation $latex 2x+5(x+1)=26$.

##### Solution

1. Simplify:

We remove parentheses: $latex 2x+5x+5=26$.

There are no fractions.

We combine like terms: $latex 7x+5=26$.

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

$latex 7x+5-5=26-5$

$latex 7x=21$

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

$latex \frac{7}{7}x=\frac{{21}}{7}$

$latex x=\frac{{21}}{7}=3$

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

$latex 2(3)+5(3+1)=26$

$latex 6+5(4)=26$

$latex 6+20=26$

This is true

**Answer:** $latex x=3$.

### EXAMPLE 4

Find the value of $latex x$ in the equation $latex \frac{1}{2}x+4(x+2)=14-2x+20$.

##### Solution

1. Simplify:

We simplify the parentheses: $latex \frac{1}{2}x+4x+8=14-2x+20$.

We simplify fractions: $latex x+8x+16=28-4x+40$.

We combine like terms: $latex 9x+16=68-4x$.

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

$latex 9x+16-16=68-4x-16$

$latex 9x=52-4x$

$latex 9x+4x=52-4x-4x$

$latex 13x=52$

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

$latex \frac{13}{13}x=\frac{{52}}{13}$

$latex x=\frac{{52}}{13}=4$

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

$latex \frac{1}{2}(4)+4(4)+8=14-2(4)+20$

$latex 2+16+8=14-8+20$

$latex 26=26$

This is true

**Answer:** $latex x=4$.

## Linear equations with one variable – Practice problems

## See also

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