Linear Equation

What is a linear equation?

An equation which involves only one variable whose highest power is 1 is known as a linear equation in that variable.

For example:

(a) x + 4 = 19

(b) y - 7 = 11

(d) 2x - 5 = x + 7

(e) x + 13 = 27

(f) y - 3 = 9

(g) 11x + 5 = x + 7

Each one of these equations is a linear equation.

The sign of equality divides the equation into two sides. Left hand side or L.H.S. and Right hand side or R.H.S

Solution of linear equation or Root of linear equation: The value of the variable which makes left hand side equal to right hand side in the given equation is called the solution or the root of the equation.

For example:

1. x + 1 = 4

Here, L.H.S. is x + 1 and R.H.S. is 4

If we put x = 3, then L.H.S. is 3 + 1 which is equal to R.H.S.

Thus, the solution of the given linear equation is x = 3

2. 5x - 2 = 3x - 4 is a linear equation.

If we put x = -1, then L.H.S. is 5 × -1 -2 and R.H.S. is 3 × -1 -4

= -5 -2 = -3 -4

= -7 = -7

So, L.H.S. = R.H.S.

Therefore, x = -1 is the solution for the equation 5x - 2 = 3x - 4

How to solve linear equation in one variable?

Rules for solving a linear equation in one variable:

The equation remains unchanged if –

(a)The same number is added to both sides of the equation.

For example:

1. x - 4 = 7

⇒ x - 4 + 4 = 7 + 4 (Add 4 to both sides)

⇒ x = 11

2. x - 2 = 10

⇒ x - 2 + 2 = 10 + 2 (Add 2 to both sides)

⇒ x = 12

(b) The same number is subtracted from both sides of the equation.

For example:

1. x + 5 = 9

⇒ x + 5 - 5 = 9 - 5 (Subtract 5 from both sides)

⇒ x + 0 = 4

⇒ x = 4

2. x + 1/2 = 3

x + 1/2 - 1/2 = 3 - 1/2 (Subtract 1/2 from both sides)

⇒ x = 3 - 1/2

⇒ x = (6 - 1)/2

⇒ x = 5/2

⇒ x/2 × 2 = 5 × 2 (Multiply 2 to both the sides)

⇒ x = 10

2. x/5 = 15

⇒ x/5 × 5 = 15/5 (Multiply 5 to both the sides)

⇒ x = 3

(d) The same non-zero number divides both sides of the equation.

For example:

1. 0.2x = 0.24

⇒ 0.2x/0.2 = 0.24/0.2 (Divide both sides by 0.2)

⇒ x = 0.12

2. 5x = 10

⇒ 5x/5 = 10/5 (Divide both sides by 2)

⇒ x = 2

What is transposition? Explain the methods of transposition.

Any term of an equation may be shifted to the other side with a change in its sign without affecting the equality. This process is called transposition.

So, by transposing a term —

● We simply change its sign and carry it to the other side of the equation.

● ‘+‘ sign of the term changes to ‘—‘ sign to the other side and vice-versa.

● ‘×’ sign of the factor changes to ‘÷‘ sign to the other side and vice-versa.

● Now, simplify L.H.S. such that each side contains just one term.

● Finally, simplify the equation to get the value of the variable.

For example:

10x - 7 = 8x + 13