In this chapter we will learn to solve linear equations with one variable. All the questions are fully solved with step by step explanation.

## Method to solve linear equation with one variable

Solving any linear equation means that we have to **find the value of given variable** which can satisfy the given equation.

When solving linear equation with one variable our main aim is to **bring the variable terms to one side** and **constant terms to other side of the equation**.

While moving terms of equation from one side to another** follow the below rules**;

(a) **Convert addition to subtraction & vice-versa**

(b) **Convert division into multiplication & vice-versa**

Division is converted into multiplication using **Cross multiplication method**

In cross multiplication, you multiply denominator of one side to numerator of other side.

For example;

\mathtt{\frac{x}{2} \ =\ x\ +\ 3}

Here number 2 is present in division form.

To move number to right side, multiply number 2 with (x+3)

As the name suggests, in cross multiplication, **we multiply denominator of one side to numerator of other side**.

This technique is very important for solving linear equations. **Conclusion**

To solve linear equation just remember the below conversion rules;

(a) ” + ” ⟹ ” – “

While moving numbers from one side to another, the addition is converted into subtraction.

Similarly;

(b) ” – ” ⟹ ” + ”

(c) ” x ” ⟹ ” ÷ ”

(d) ” ÷ ” ⟹ ” x “

## Solving Linear Equations – Solved Examples

**Example 01**

Solve 2x + 9 = 21

**Solution**

We separate variable x on one side and constant on other side.

**Move 9 to right**.

The addition will convert into subtraction.

2x + 9 = 21

2x = 21- 9

2x = 12**Move 2 towards right**.

The multiplication will convert into division.

2x = 12

x = 12 / 2

x = 6

Hence, **the value of x is 6**.

**Example 02**

Find the value of x in given linear equation

2 (6x – 3) + 4 = 3x + 7

**Solution**

First simplify the given equation by removing bracket.

In equations, the removal of bracket results in multiplication of numbers.

2.6x – 2.3 + 4 = 3x + 7

12x – 6 + 4 = 3x + 7

Now we have to separate variable x and constants on different sides.

**Move 3x towards left**.

Move 6 & 4 towards right.

12x – 6 + 4 = 3x + 7

12x – 3x = 7 + 6 – 4

9x = 9**Move 9 towards right.**

The multiplication will convert into division.

9x = 9

x = 9 / 9

x = 1

Hence, **for given linear equation the value of x is 1**.

**Example 03**

Solve the given linear equation

\mathtt{\frac{x-3}{4} \ \ =\ \frac{2x}{5}}

**Solution**

Cross multiply number 4 & 5 to the right and left side respectively.

\mathtt{\frac{x-3}{4} \ \ =\ \frac{2x}{5}}\\\ \\ \mathtt{5\ ( \ x\ -\ 3) \ =\ 4.\ 2x}\\\ \\ \mathtt{5x\ -\ 15\ =\ 8x}

Now separate variables on one side and constant on the other side.**Move 15 to the right**.**Move 8x to the left**.

5x – 15 = 8x

5x – 8x = 15

-3x = 15

**Move -3 to the right**.

The multiplication will convert into division.

-3x = 15

x = 15 / -3

x = -5

Hence, in given linear equation, **the value of x is -5**.

**Example 04**

Solve the below linear equation.

\mathtt{\frac{4x-3}{5} \ -6x\ =\ \frac{2x}{5}}

**Solution**

First simplify the left side of equation.

\mathtt{\frac{4x-3-30x}{5} =\ \frac{2x}{5}}

Since both side of equation contain same denominator, we can cancel out each of them.

\mathtt{\frac{4x-3-30x}{\cancel{5}} =\ \frac{2x}{\cancel{5}}}\\\ \\ \mathtt{4x\ -3-30x\ =\ 2x}

Separate variables on left side and constant on right side.**Move 2x to the left and 3 to the right**.

4x – 3 – 30x = 2x

4x – 30x – 2x = 3

-28x = 3

**Move -28 towards right.**

Multiplication will convert into division.

x = – 3 / 28

Hence, for given linear equation, the **value of x is -3 / 28.**

**Example 05**

4(3x + 3) – 5( 4 – 7x ) = 2(x + 2) + 5( x + 9)

**Solution**

Simplify the equation.

Remove the brackets and convert into multiplication.

4.3x + 4.3 – 5.4 + 5.7x = 2.x + 2.2 + 5.x + 5.9

12x + 12 – 20 + 35x = 2x + 4 + 5x + 45

12x + 35x + 12 – 20 = 2x + 5x + 4 + 45

47x – 8 = 7x + 49

Separate variable x on left and constants on right.**Move 8 towards right**.**Move 7x towards left.**

47x – 8 = 7x + 49

47x – 7x = 49 + 8

40x = 57**Move 40 towards right.**

Multiplication will convert into division.

40x = 57

x = 57 / 40

Hence, **the value of x is 57 / 40** for the given algebraic expression.

**Example 06**

Solve the below algebraic expression.

\mathtt{\frac{x}{2} -\frac{x}{5} =\ 2x-\ 9}

**Solution**

First simplify the expression.

\mathtt{\frac{x}{2} -\frac{x}{5} =\ 2x-\ 9}\\\ \\ \mathtt{\frac{5x-2x}{10} =\ 2x\ -\ 9}\\\ \\ \mathtt{\frac{3x}{10} =\ 2x\ -\ 9}

**Cross multiply the denominator 10 to the right side**.

\mathtt{3x\ =\ 10\ ( 2x\ -\ 9)}\\\ \\ \mathtt{3x\ =\ 20x\ -90}

**Move 20x towards left.**

3x – 20x = -90

-17x = -90**Take -17 into right side of equation**.

The multiplication will convert into division.

x = -90 / -17

x = 90 / 17

Hence for given algebraic expression **the value of x is 90 / 17**.