# Solving linear equations

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.