# 20 Linear Equation Examples with Answers

Linear equations can be solved by applying various operations to both sides of the equal sign. These operations can help us simplify the equation, solve for the variable, and ultimately find the solution.

In this article, we will look at a brief summary of linear equations, followed by 20 examples with answers to master the process of solving first-degree equations.

##### ALGEBRA

Relevant for

Learning to solve linear equations with examples.

See examples

##### ALGEBRA

Relevant for

Learning to solve linear equations with examples.

See examples

## How to solve linear equations?

Recall that linear equations are equations in which all variables have a maximum power of 1. For example, the equations $latex 4x+1=5$ and $latex 2x+12=4x-2$ are linear equations.

To solve linear equations, we have to apply different operations to both sides of the equal sign, so that we can solve for the variable. Therefore, we can follow the following steps to find the solution to linear equations:

Step 1: We simplify the expression. This includes removing parentheses and other grouping signs, removing fractions, and combining like terms.

Step 2: We isolate the variable. We perform addition and subtraction to place all terms with variables on only one side of the equation.

Step 3: We solve the equation. We do multiplication and division to find the answer.

## 20 Linear equation examples with answers

The following 20 linear equation examples have their respective solution, where the process is indicated step by step. It is recommended that you try to solve the examples yourself before looking at the answer.

### EXAMPLE 1

Solve the equation $latex 5x-12=3$.

Step 1: Simplify: We have nothing to simplify here.

Step 2: Solve for the variable: We use addition to solve for the variable:

$latex 5x-12=3$

$latex 5x-12+12=3+12$

$latex 5x=15$

Step 3: Solve: We divide both sides by 5:

$$\frac{5x}{5}=\frac{15}{5}$$

$latex x=3$

### EXAMPLE 2

Solve the equation $latex 3x+1=x-3$.

Step 1: Simplify: We have nothing to simplify.

Step 2: Solve for the variable: We use addition and subtraction to solve for the variable:

$latex 3x+1=x-3$

$latex 3x+1-1=x-3-1$

$latex 3x=x-4$

$latex 3x-x=x-4-x$

$latex 2x=-4$

Step 3: Solve: We divide both sides by 2:

$$\frac{2x}{2}=\frac{-4}{2}$$

$latex x=-2$

### EXAMPLE 3

Find the value of t in the equation $latex 5t+5=3t+7$.

Step 1: Simplify: We do not have like terms.

Step 2: Solve for the variable: We use subtraction to solve for the variable:

$latex 5t+5=3t+7$

$latex 5t+5-5=3t+7-5$

$latex 5t=3t+2$

$latex 5t-3t=3t+2-3t$

$latex 2t=2$

Step 3: Solve: We divide both sides by 2:

$$\frac{2t}{2}=\frac{2}{2}$$

$latex t=1$

### EXAMPLE 4

Solve the equation $latex 3(2x+1)=-9$.

Step 1: Simplify: We expand the parentheses:

$latex 3(2x+1)=-9$

$latex 6x+3=-9$

Step 2: Solve for the variable: We use subtraction to solve for the variable:

$latex 6x+3=-9$

$latex 6x+3-3=-9-3$

$latex 6x=-12$

Step 3: Solve: We divide both sides by 6:

$$\frac{6x}{6}=\frac{-12}{6}$$

$latex x=-2$

### EXAMPLE 5

Solve the equation $latex 2(2x-5)=3(x-1)-4$.

Step 1: Simplify: We expand the parentheses on both sides of the equation and combine like terms:

$latex 2(2x-5)=3(x-1)-4$

$latex 4x-10=3x-3-4$

$latex 4x-10=3x-7$

Step 2: Solve for the variable: We use addition and subtraction to solve for the variable:

$latex 4x-10+10=3x-7+10$

$latex 4x=3x+3$

$latex 4x-3x=3-3x$

$latex x=3$

Step 3: Solve: In this case, we no longer have to divide:

$latex x=3$

### EXAMPLE 6

Find the value of z in the equation $latex 3(z-2)+10=2(2z+2)+2$.

Step 1: Simplify: We expand the parentheses and combine like terms:

$latex 3(z-2)+10=2(2z+2)+2$

$latex 3z-6+10=4z+4+2$

$latex 3z+4=4z+6$

Step 2: Solve for the variable: We use subtraction to solve for the variable:

$latex 3z+4-4=4z+6-4$

$latex 3z=4z+2$

$latex 3z-4z=2$

$latex -z=2$

Step 3: Solve: We divide both sides by -1:

$$\frac{-z}{-1}=\frac{2}{-1}$$

$latex z=-2$

### EXAMPLE 7

Solve the equation $latex \frac{2x+1}{3}=x-1$.

Step 1: Simplify: We multiply by 3 to eliminate the fraction:

$$\frac{2x+1}{3}=x-1$$

$latex 2x+1=3x-3$

Step 2: Solve for the variable: We subtract 1 and 3x from both sides:

$latex 2x+1=3x-3$

$latex 2x+1-1=3x-3-1$

$latex 2x=3x-4$

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

$latex -x=-4$

Step 3: Solve: We divide both sides by -1:

$$\frac{-x}{-1}=\frac{-4}{-1}$$

$latex x=4$

### EXAMPLE 8

Solve the equation $latex \frac{4x}{3}-2= \frac{2x+3}{3} -1$.

Step 1: Simplify: We multiply both sides of the equation by 3 to eliminate the fractions and combine like terms:

$$\frac{4x}{3}-2=\frac{2x+3}{3}-1$$

$latex 4x-6=2x+3-3$

$latex 4x-6=2x$

Step 2: Solve for the variable: We add 6 and subtract 2x from both sides:

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

$latex 4x=2x+6$

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

$latex 2x=6$

Step 3: Solve: We divide both sides by 2:

$$\frac{2x}{2}=\frac{6}{2}$$

$latex x=3$

### EXAMPLE 9

Find the value of t in the equation $latex \frac{2t-5}{5}+2=\frac{t-2}{3}+2$.

Step 1: Simplify: We multiply by 15 to eliminate the fractions and combine like terms:

$$\frac{2t-5}{5}+2=\frac{t-2}{3}+2$$

$$3(2t-5)+15(2)=5(t-2)+15(2)$$

$latex 6t-15+30=5t-10+30$

$latex 6t+15=5t+20$

Step 2: Solve for the variable: We subtract 15 and 5 t from both sides:

$latex 6t+15=5t+20$

$latex 6t+15-15=5t+20-15$

$latex 6t=5t+5$

$latex 6t-5t=5t+5-5t$

$latex t=5$

Step 3: Solve: We no longer have to divide:

$latex t=5$

### EXAMPLE 10

Solve the equation $latex \frac{2x-3}{x+1}+2=3$.

Step 1: Simplify: We multiply both sides by (x+1) and combine like terms:

$$\frac{2x-3}{x+1}+2=3$$

$latex 2x-3+2(x+1)=3(x+1)$

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

$latex 4x-1=3x+3$

Step 2: Solve for the variable: Add 1 and subtract 3 x from both sides:

$latex 4x-1=3x+3$

$latex 4x-1+1=3x+3+1$

$latex 4x=3x+4$

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

$latex x=4$

Step 3: Solve: We no longer have to divide:

$latex x=4$

### EXAMPLE 11

Find the value of t in the equation $latex 3t+4(t-10)=t+20$.

Step 1: Simplify: We expand the parentheses and combine like terms:

$latex 3t+4(t-10)=t+20$

$latex 3t+4t-40=t+20$

$latex 7t-40=t+20$

Step 2: Solve for the variable: We add 40 and subtract t from both sides:

$latex 7t-40=t+20$

$latex 7t-40+40=t+20+40$

$latex 7t=t+60$

$latex 7t-t=t+60-t$

$latex 6t=60$

Step 3: Solve: We divide both sides by 6:

$$\frac{6t}{6}=\frac{60}{6}$$

$latex t=10$

### EXAMPLE 12

Solve the equation $latex 3x+6(x+1)=3(x+1)+5$.

Step 1: Simplify: We expand the parentheses and combine like terms:

$latex 3x+6(x+1)=3(x+1)+5$

$latex 3x+6x+6=3x+3+5$

$latex 9x+6=3x+8$

Step 2: Solve for the variable: We subtract 6 and 3 x from both sides:

$latex 9x+6=3x+8$

$latex 9x+6-6=3x+8-6$

$latex 9x=3x+2$

$latex 9x-3x=3x+2-3x$

$latex 6x=2$

Step 3: Solve: We divide both sides by 6:

$$\frac{6x}{6}=\frac{2}{6}$$

$$x=\frac{1}{3}$$

### EXAMPLE 13

Find the value of x in the equation $latex \frac{1}{x+2}+2=\frac{9}{4}$.

Step 1: Simplify: We multiply the entire equation by 4 (x+2) and combine like terms:

$$\frac{1}{x+2}+2=\frac{9}{4}$$

$latex 4+8(x+2)=9(x+2)$

$latex 4+8x+16=9x+18$

$latex 8x+20=9x+18$

Step 2: Solve for the variable: We subtract 20 and 9 x from both sides:

$latex 8x+20-20=9x+18-20$

$latex 8x=9x-2$

$latex 8x-9x=9x-2-9x$

$latex -x=-2$

Step 3: Solve: We divide both sides by -1:

$$\frac{-x}{-1}=\frac{-2}{-1}$$

$latex x=2$

### EXAMPLE 14

Find the value of y in the equation $$2y+3(2y-5)+4=y+3(2y-2)-6$$

Step 1: Simplify: We expand the parentheses and combine like terms:

$$2y+3(2y-5)+4=y+3(2y-2)-6$$

$latex 2y+6y-15+4=y+6y-6-6$

$latex 8y-11=7y-12$

Step 2: Solve for the variable: We add 11 and subtract 7y from both sides:

$latex 8y-11=7y-12$

$latex 8y-11+11=7y-12+11$

$latex 8y=7y-1$

$latex 8y-7y=7y-1-7y$

$latex y=-1$

Step 3: Solve: We no longer have to divide:

$latex y=-1$

### EXAMPLE 15

Solve the equation $latex \frac{4x-9}{3}+2=3(x-2)$.

Step 1: Simplify: We multiply the entire equation by 3, expand the parentheses, and combine like terms:

$$\frac{4x-9}{3}+2=3(x-2)$$

$latex 4x-9+6=9(x-2)$

$latex 4x-3=9x-18$

Step 2: Solve for the variable: We add 3 and subtract 9 x from both sides:

$latex 4x-3+3=9x-18+3$

$latex 4x=9x-15$

$latex 4x-9x=9x-15-9x$

$latex -5x=-15$

Step 3: Solve: We divide both sides by -5:

$$\frac{-5x}{-5}=\frac{-15}{-5}$$

$latex x=3$

### EXAMPLE 16

Find the value of x in the equation $latex -3x+18=-x(13-10)+4x-2$.

Step 1: Simplify: We expand the parentheses and combine like terms:

$latex -3x+18=-x(13-10)+4x-2$

$latex -3x+18=-x(3)+4x-2$

$latex -3x+18=-3x+4x-2$

$latex -3x+18=x-2$

Step 2: Solve for the variable: We subtract 18 and subtract x from both sides:

$latex -3x+18-18=x-2-18$

$latex -3x=x-20$

$latex -3x-x=x-20-x$

$latex -4x=-20$

Step 3: Solve: We divide both sides by -4:

$$\frac{-4x}{-4}=\frac{-20}{-4}$$

$latex x=5$

### EXAMPLE 17

Find the value of w in the equation $latex 10(2w-5)=2w+2(w+1)$.

Step 1: Simplify: We expand the parentheses and combine like terms:

$latex 10(2w-5)=2w+2(w+1)$

$latex 20w-50=2w+2w+2$

$latex 20w-50=4w+2$

Step 2: Solve for the variable: We add 50 and subtract 4 w from both sides:

$latex 20w-50=4w+2$

$latex 20w-50+50=4w+2+50$

$latex 20w=4w+52$

$latex 20w-4w=4w+52-4w$

$latex 16w=52$

Step 3: Solve: We divide both sides by 16 and simplify the fraction:

$$\frac{16w}{16}=\frac{52}{16}$$

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

### EXAMPLE 18

Find the value of r in the equation $latex 3(-2r-5)+4=\frac{r}{2}+2$.

Step 1: Simplify: We multiply both sides by 2 to eliminate the fraction, expand the parentheses, and combine like terms:

$$3(-2r-5)+4=\frac{r}{2}+2$$

$latex 6(-2r-5)+8=r+4$

$latex -12r-30+8=r+4$

$latex -12r-22=r+4$

Step 2: Solve for the variable: We add 22 and subtract r from both sides:

$latex -12r-22=r+4$

$latex -12r-22+22=r+4+22$

$latex -12r=r+26$

$latex -12r-r=r+26-r$

$latex -13r=26$

Step 3: Solve: We divide both sides by -13:

$$\frac{-13r}{-13}=\frac{26}{-13}$$

$latex x=-2$

### EXAMPLE 19

Find the value of x in the equation $$3x+4(-2x+1)=3(x-5)+2(2x-7)-3$$

Step 1: Simplify: We expand all the parentheses and combine like terms:

$$3x+4(-2x+1)=3(x-5)+2(2x-7)-3$$

$$3x-8x+4=3x-15+4x-14-3$$

$latex -5x+4=7x-32$

Step 2: Solve for the variable: We subtract 4 and 7 x from both sides:

$latex -5x+4-4=7x-32-4$

$latex -5x=7x-36$

$latex -5x-7x=7x-36-7x$

$latex -12x=-36$

Step 3: Solve: We divide both sides by -12:

$$\frac{-12x}{-12}=\frac{-36}{-12}$$

$latex x=3$

### EXAMPLE 20

Find the value of x in the equation $latex 2\left(\frac{x+2}{4} \right)+2=\frac{3x}{4}+2$.

Step 1: Simplify: We start by simplifying the fraction, then multiply by 4 to eliminate the fractions and combine like terms:

$$2\left( \frac{x+2}{4}\right)+2=\frac{3x}{4}+2$$

$$\frac{x+2}{2}+2=\frac{3x}{4}+2$$

$latex 2(x+2)+8=3x+8$

$latex 2x+4+8=3x+8$

$latex 2x+12=3x+8$

Step 2: Solve for the variable: We subtract 12 and 3 x from both sides:

$latex 2x+12-12=3x+8-12$

$latex 2x=3x-4$

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

$latex -x=-4$

Step 3: Solve: We divide both sides by -1:

$$\frac{-x}{-1}=\frac{-4}{-1}$$

$latex x=4$