Linear equations can be solved by applying various operations to both sides of the equal sign so that we completely solve for the variable. This is possible because if we perform an operation on both sides of the equation, we are not actually changing the equation.

In this article, we will look at a summary of linear equations and explore various exercises in linear equations solved to master the process of solving these equations.

## Summary of linear equations

Linear equations, also known as first-degree equations, are equations that have variables with a maximum power of 1. For example, the equations $latex 5x-4=9$ and $latex 4x-4=x-5$ are linear equations with one unknown. Linear equations with one unknown can be solved by following the following steps:

**Step 1:** Simplify: We simplify the given equation to facilitate its resolution. This includes removing parentheses and other grouping signs, removing fractions, and combining like terms

** Step 2:** Solve for the variable: We use addition and subtraction to move all the variables to one side of the equation and the constant terms to the other.

** Step 3:** Solve: We use division or multiplication to completely isolate the variable and obtain the solution.

## Exercises with answers of linear equations

The following linear equations exercises are solved by following the steps detailed above. These exercises can be used to fully master the topic of linear equations.

**EXERCISE 1**

What is the value of *x* in the linear equation $latex 5x-10=10$?

##### Solution

**Step 1: **Simplify: We do not have parentheses, fractions, or like terms.

**Step 2:** Solve for the variable: Add 10 to both sides of the equation:

$latex 5x-10=10$

$latex 5x-10+10=10+10$

$latex 5x=20$

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

$latex \frac{5x}{5}=\frac{20}{5}$

$latex x=4$

**EXERCISE 2**

What is the value of *y* in the linear equation $latex 4y-5=3y+1$?

##### Solution

**Step 1: **Simplify: We do not have parentheses, fractions, or like terms.

**Step 2:** Solve for the variable: We add 5 and subtract 3*y* from both sides:

$latex 4y-5=3y+1$

$latex 4y-5+5=3y+1+5$

$latex 4y=3y+6$

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

$latex y=6$

**Step 3:** Solve: We have already got the answer:

$latex y=6$

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**EXERCISE 3**

Find the value of *x* in the linear equation $latex 2(2x-3)=2x-10$.

##### Solution

**Step 1: **Simplify: We expand the parentheses:

$latex 2(2x-3)=2x-10$

$latex 4x-6=2x-10$

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

$latex 4x-6+6=2x-10+6$

$latex 4x=2x-4$

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

$latex 2x=-4$

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

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

$latex x=-2$

**EXERCISE 4**

Solve the linear equation $latex 3x-4=3(2x-2)-7$.

##### Solution

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

$latex 3x-4=3(2x-2)-7$

$latex 3x-4=6x-6-7$

$latex 3x-4=6x-13$

**Step 2:** Solve for the variable: We add 4 and subtract 6*x* from both sides:

$latex 3x-4+4=6x-13+4$

$latex 3x=6x-9$

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

$latex -3x=-9$

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

$latex \frac{-3x}{-3}=\frac{-9}{-3}$

$latex x=3$

**EXERCISE 5**

Find the value of *t* in the linear equation $latex 2(t+2)-5=5(t-4)+13$.

##### Solution

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

$latex 2(t+2)-5=5(t-4)+13$

$latex 2t+4-5=5t-20+13$

$latex 2t-1=5t-7$

**Step 2:** Solve for the variable: We add 1 and subtract 5*t* from both sides:

$latex 2t-1+1=5t-7+1$

$latex 2t=5t-6$

$latex 2t-5t=5t-6-5t$

$latex -3t=-6$

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

$latex \frac{-3t}{-3}=\frac{-6}{-3}$

$latex x=2$

**EXERCISE 6**

What is the value of *x* in the equation $latex \frac{3x}{2}+4=2x+5$?

##### Solution

**Step 1: **Simplify: We multiply the entire equation by 2 to eliminate the fraction:

$latex \frac{3x}{2}+4=2x+5$

$latex 3x+4(2)=2x(2)+5(2)$

$latex 3x+8=4x+10$

**Step 2:** Solve for the variable: We subtract 8 and 4*x* from both sides:

$latex 3x+8-8=4x+10-8$

$latex 3x=4x+2$

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

$latex -x=2$

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

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

$latex x=-2$

**EXERCISE 7**

Find the value of *x* in the equation $latex \frac{x+6}{3}+4=\frac{3x-5}{2}+2x-1$.

##### Solution

**Step 1: **Simplify: We multiply the entire equation by 6 to eliminate the fractions and combine like terms:

$latex \frac{x+6}{3}+4=\frac{3x-5}{2}+2x-1$

$$2(x+6)+4(6)=3(3x-5)+2x(6)-1(6)$$

$$2x+12+24=9x-15+12x-6$$

$latex 2x+36=21x-21$

**Step 2:** Solve for the variable: We subtract 36 and 21*x* from both sides:

$latex 2x+36-36=21x-21-36$

$latex 2x=21x-57$

$latex 2x-21x=21x-57-21x$

$latex -19x=-57$

**Step 3:** Solve: We divide both sides by -19:

$latex \frac{-19x}{-19}=\frac{-57}{-19}$

$latex x=3$

## Linear equations – Exercises to solve

Practice solving linear equations with the following exercises. Simply select your answer and verify it by clicking “Check”. You can look at the solved exercises above if you have problems with these exercises.

## See also

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

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