**Notes of chapter: Linear Equations in One Variable are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 8.**

**(1)Variable-**

A **variable **takes on different numerical values; i.e., its value is not fixed. Variables are usually denoted by alphabets, such as x, y, z, m, n, o etc.

**(2**)**Coefficient-**

** Coefficient** of the variables is a number that is multiplied with the variables.

**Eg :-** 3x

3 is multiplied to x, therefore, 3 is coefficient of variable x.

**(3) Operator-**

The sign of mathematical operation (+, _ ,× , ÷) is known as **operator**.

**(4) Constant-**

A **constant** is a mathematical value which can not be changed in any circumstances. A number is

**Eg:-** 6, 3, 16 etc. are constant values.

**(5) Terms-**

A single number (4) or a variable (x) or a number multiplied with variable (4x) is known as **terms**.

**Eg:-**

3x + 7 = 10

In above equation

3x, 7 and 10 all are terms.

**(7)Equation –**

An** equation** is a condition on a variable.

The condition is that two expressions should have equal value, i.e., they should have **equal sign** between them. The sign of equality shows that the value of the expression to the left hand side of the sign (LHS) is equal to the value of the expression to the right side of the sign(RHS).

At least one expression of them should have variable.

**Eg:-**

4x + 3 = 15

The condition on the variable x states that the value of (4x + 3)is 15.

Therefore,

4x + 3 = 15

If x = 1

4 1 + 3 = 15

4 + 3 =15

7 ≠ 15

LHS ≠ RHS

But if x = 3

4 × 3 + 3 = 15

12+ 3 =15

15 = 15

LHS = RHS

Hence, x =3 is the only solution for the variable x. No other value of x =1, x=2 etc. can satisfies the condition of equation. Therefore, x = 3 satisfies the condition of equation 4x + 3 = 15.

**(8)Solving an Equation-**

**(i)Solving equations by different mathematical operations –**

Equation should be equal at both sides when we apply any operation of addition, subtraction, multiplication or division. Therefore, we should **apply operation both sides of equation.**

** (a)Solving equation by addition-**

x – 2 = 4

Adding 2 both sides [to maintain equality of the equation]

x – 2 + 2 = 4 + 2

x + 0 = 6

x = 6

**(b) Solving equation by subtraction-**

x + 2 = 4

Subtracting 2 both sides [to maintain equality of the equation]

x + 2 – 2 = 4 – 2

x + 0 = 2

x = 2

**(c) Solving equation by multiplication-**

x = 4

Multiply 1 both sides [to maintain equality of the equation]

x × 1 = 4×1

x = 4

**(ii)Solving equation by transposing** **a number** –

Transferring a number or a term from one side of equation to other side is called **transposition**. While transferring a number its sign should be change.

If +3 is transposed to other side, it will be written with – sign,ie, -3.

If -3 is transposed to other side, it will be written with + sign,ie, +3.

**Eg :-**

**(i) **x + 3 = 8

Transposing 3 from LHS to RHS with negative sign

x = 8 – 3

x = 5

**(ii)** x – 4 = 7

Transposing 4 from LHS to RHS with positive sign

x = 7 + 4

x = 11

**(9)From solution to equation-**

To form an equation from solution, we will apply different operation of mathematics on both sides of the solution. We can make many equations from the solution but we cannot make many solutions from one equation.

**Eg:-** Let solution is

x = 2

**Equation 1** from given solution

Add 2 both sides

x + 2 = 2 + 2

x + 2 = 4

**Equation 2** from given solution

x = 2

Subtract 2 from both sides

x – 2 = 2 – 2

x – 2 = 0

**Equation 3** from given solution

x= 2

Multiply 2 both sides

2x = 2×2

2x = 4

Add 8 both sides

2x + 8 = 4 + 8

2x + 8 = 12

**(10)Linear equations-**

** Linear equations **are those equations or expressions which have the highest power of the variable appearing in the expression is 1.

**Eg:-**

2x, 2x + 1

**(11)Linear equation in one variable**

The linear equation with one variable is called **linear equation in one variable.**

**Eg:-**

3y-5

In above equation,

y is a variable.

5 is constant.

3 is coefficient of variable y.

**(12)Solving equations having the variable on both sides-**

**Eg:-** Solve 2x – 3 = x + 2

Solution

2x – 3 = x + 2

**Step 1-**

Transpose variables to one side of the equation and constants on the other side of the equation.

Transposing 3 to RHS and x to LHS

2x – x = 2 + 3

x = 5

**Helping Topics**