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 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:** Isolate 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

**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:** Isolate 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:

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

$latex x=4$

**EXERCISE **2

Find the value of *x* in the equation

$$2x-4+2x=12$$

##### Solution

**Step 1:** Simplify: We can combine like terms on the left-hand side of the equation:

$latex 2x-4+2x=12$

$latex 4x-4=12$

**Step 2:** Isolate the variable: We add 4 to both sides:

$latex 4x-4=12$

$latex 4x-4+4=12+4$

$latex 4x=16$

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

$$\frac{4x}{4}=\frac{16}{4}$$

$latex x=4$

**EXERCISE **3

**EXERCISE**

Solve the following linear equation:

$$4t-6=10-4t$$

##### Solution

**Step 1: **Simplify: We cannot combine like terms initially.

**Step 2:** Solve for the variable: Add 6 and 4t on both sides of the equation:

$latex 4t-6=10-4t$

$latex 4t-6+6=10-4t+6$

$latex 4t=16-4t$

$latex 4t+4t=16-4t+4t$

$latex 8t=16$

**Step 3:** Solve: Both sides of the equation are divided by 8.:

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

$latex x=3$

**EXERCISE **4

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:** Isolate 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:

$latex y=6$

**EXERCISE **5

Find the value of *x* in the linear equation

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

##### Solution

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

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

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

** Step 2:** Isolate 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:

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

$latex x=-2$

**EXERCISE **6

Solve the linear equation

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

##### 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:** Isolate 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:

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

$latex x=3$

**EXERCISE **7

**EXERCISE**

Find the solution to the following equation:

$$2t-6+3t=2(-2t+2)+8$$

##### Solution

**Step 1: **Simplify: Use the distributive property for the parenthesis and combine like terms.:

$$2t-6+3t=2(-2t+2)+8$$

$$5t-6=2(-2t+2)+8$$

$$5t-6=-4t+4+8$$

$$5t-6=-4t+12$$

**Step 2:** Solve for the variable: Add 6 and 4t on both sides of the equation:

$$5t-6+6=-4t+12+6$$

$latex 5t=-4t+18$

$latex 5t+4t=-4t+18+4t$

$latex 9t=18$

**Step 3:** Solve: Divide both sides by 9:

$$\frac{9t}{9}=\frac{18}{9}$$

$latex x=2$

**EXERCISE **8

Find the value of *t* in the linear equation

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

##### 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:** Isolate 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:

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

$latex x=2$

**EXERCISE **9

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:

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

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

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

** Step 2:** Isolate 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:

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

$latex x=-2$

**EXERCISE **10

Find the value of *x* in the equation

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

##### Solution

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

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

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

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

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

** Step 2:** Isolate 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:

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

$latex x=3$

## Linear equations – Exercises to solve

#### Solve the linear equation $latex 4(x+5)=-5x-7$.

Write the answer in the input box.

## See also

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