Solution of a linear equation

Solution of a linear equation

Worked-out examples on solving linear equations are given below. Instructions are given step-by-step with detailed explanation by using addition, subtraction, multiplication and division for solving linear equations.

Examples on Solving Linear Equations:

1. Solve: (2x + 5)/(x + 4) = 1

Solution:

(2x + 5)/(x + 4) = 1

⇒ 2x + 5 = 1(x + 4) 

⇒ 2x + 5 = x + 4 

⇒ 2x - x = 4 - 5   (Transferring positive x to the left hand side changes to negative x and again, positive 5 changes to negative 5) 

⇒ x = -1 

Therefore, x = - 1 is the required solution of the equation (2x + 5)/(x + 4) = 1 

2. Solve: 6x - 19 = 3x - 10

Solution:

6x - 19 = 3x - 10

⇒ 6x - 3x = - 10 + 19   (Transferring 3x to L.H.S changes to negative 3x and -19 to R.H.S. changes to positive 19)

⇒ 3x = 9

⇒ 3x/3 = 9/3   (Dividing both sides by 3)

⇒ x = 3

3. Solve: 5 - 2(x - 1) = 4(3 - x) - 2x.

Solution:

5 - 2(x - 1) = 4(3 - x) - 2x

⇒ 5 - 2x + 2 = 12 - 4x - 2x   (Removing the brackets and then simplify)

⇒ 7 - 2x = 12 - 6x   (Transferring -6x to L.H.S. changes to positive 6x and 7 to R.H.S. changes to negative 7)

⇒ -2x + 6x = 12 - 7

⇒ 4x = 5

⇒ 4x/4 = 5/4

⇒ x = 5/4

4. x/2 + x/3 = x - 7

Solution:

x/2 + x/3 = x - 7

Least common multiple of2 and 3 is 6

⇒ (x/2 × 3/3) + (x/3 × 2/2) = x - 7

⇒ 3x/6 + 2x/6 = x - 7

⇒ (3x + 2x)/6 = x - 7

⇒ 5x/6 = x - 7

⇒ 5x = 6(x - 7)

⇒ 5x = 6x - 42   (Transferring 6x to L.H.S. changes to negative 6x)

⇒ 5x - 6x = -42

⇒ -x = -42

⇒ x = 42

5. 2/(x + 3) = 3/(5 - x)

Solution:

2/(x + 3) = 3/(5 - x)

⇒ 3(x + 3) = 2(5 - x)   (cross multiply and then remove the brackets)

⇒ 3x + 9 = 10 - 2x   (Transferring -2x to L.H.S. changes to positive 2x and 9 to R.H.S. changes to -9)

⇒ 3x + 2x = 10 - 9

⇒ 5x = 1

⇒ 5x/5 = 1/5   (Dividing both sides by 5)

⇒ x = 1/5

Solution of a linear equation

Worked-out examples on solving linear equations are given below. Instructions are given step-by-step with detailed explanation by using addition, subtraction, multiplication and division for solving linear equations.

Examples on Solving Linear Equations:

1. Solve: (2x + 5)/(x + 4) = 1

Solution:

(2x + 5)/(x + 4) = 1
⇒ 2x + 5 = 1(x + 4) 
⇒ 2x + 5 = x + 4 
⇒ 2x - x = 4 - 5   (Transferring positive x to the left hand side changes to negative x and again, positive 5 changes to negative 5) 

⇒ x = -1 
Therefore, x = - 1 is the required solution of the equation (2x + 5)/(x + 4) = 1 

2. Solve: 6x - 19 = 3x - 10

Solution:

6x - 19 = 3x - 10
⇒ 6x - 3x = - 10 + 19   (Transferring 3x to L.H.S changes to negative 3x and -19 to R.H.S. changes to positive 19)

⇒ 3x = 9
⇒ 3x/3 = 9/3   (Dividing both sides by 3)
⇒ x = 3
3. Solve: 5 - 2(x - 1) = 4(3 - x) - 2x.

Solution:

5 - 2(x - 1) = 4(3 - x) - 2x
⇒ 5 - 2x + 2 = 12 - 4x - 2x   (Removing the brackets and then simplify)

⇒ 7 - 2x = 12 - 6x   (Transferring -6x to L.H.S. changes to positive 6x and 7 to R.H.S. changes to negative 7)

⇒ -2x + 6x = 12 - 7
⇒ 4x = 5
⇒ 4x/4 = 5/4
⇒ x = 5/4
4. x/2 + x/3 = x - 7

Solution:

x/2 + x/3 = x - 7
Least common multiple of2 and 3 is 6
⇒ (x/2 × 3/3) + (x/3 × 2/2) = x - 7
⇒ 3x/6 + 2x/6 = x - 7
⇒ (3x + 2x)/6 = x - 7
⇒ 5x/6 = x - 7
⇒ 5x = 6(x - 7)
⇒ 5x = 6x - 42   (Transferring 6x to L.H.S. changes to negative 6x)

⇒ 5x - 6x = -42
⇒ -x = -42
⇒ x = 42
5. 2/(x + 3) = 3/(5 - x)

Solution:

2/(x + 3) = 3/(5 - x)
⇒ 3(x + 3) = 2(5 - x)   (cross multiply and then remove the brackets)

⇒ 3x + 9 = 10 - 2x   (Transferring -2x to L.H.S. changes to positive 2x and 9 to R.H.S. changes to -9)

⇒ 3x + 2x = 10 - 9
⇒ 5x = 1
⇒ 5x/5 = 1/5   (Dividing both sides by 5)
⇒ x = 1/5