# CLASS-8INTRODUCTION OF LINEAR EQUATIONS

Solving Linear Equations

To solve linear equations we need to follow step by step which are given below

Step.1) Simplify both sides of the equation, we have to use distributive law (if necessary) to separate terms containing the variable and the constant terms.

Step.2) If the equation involve fractions multiply both sides by the LCM of the denominators to clear the fractions.

Step.3) If decimals are present, multiply both sides by a suitable power of 10 to eliminate the decimals

Step.4) Collect all the terms containing the variable on one side of the equation (generally, LHS) and all constant terms on the other side.

Step.5) Divide both sides of the equation by the resulting coefficient of the variable.

Verification of the Solution -

Substitute the value of the variable on both sides of the equation, if the values of both sides are equal the solution of the equation is correct.

There are some example are given below for your better understanding  –

Example.1) Solve 3 – 2 (4x + 5) = 8 – 3x and verify the solution

Ans.)  The equation  3 – 2 (4x + 5) =  8 – 3x

Simplifying,  3 – 8x – 10 = 8 – 3x

- 8x - 7 = 8 – 3x ;  - 8x + 3x = 8 + 7 ;  - 5x = 15 ;  x  =  - 3

Verification,

Substituting  x = - 3 on the LHS , we get

3 – 2 (4x + 5)

= 3 – 8x – 10 =  3 – 8 X ( -3) – 10  = 24 – 7  = 17

Substituting  x = - 3 on the RHS , we get

8 – 3x = 8 – 3 X (-3)  = 8 + 9 = 17

So, we can observe that LHS = RHS

So, we can conclude that, the solution x = - 3 is correct.   (Ans.)

Example.2)  Solve 18 – 3 (8x + 2) = 8 (5 - 2x) – 4x and verify the solution

Ans.)  The given equation 8 – 3 (8x + 2) =  8 (5 – 2x) – 4x

Simplifying, 18 – 3 (8x + 2) =  8 (5 - 2x) – 4x

18 – 24x – 6 =  40 – 16x – 4x

Or,   12 – 40  =  24x – 20x

Or,   - 28 = 4x

Or,    x =  - 7

Verification,

Substituting  x = - 7 on the LHS , we get

18 – 3 (8x + 2) = 18 – 3 { 8 X (-7) + 2}

= 18 – 3 (-56 + 2)

=  18 – 3 X (- 54)

=  18 + 162 =  180

Substituting  x = - 7 on the RHS, we get

8 (5 – 2x) – 4x = 40 – 16x – 4x

= 40 – 20x

= 40 – 20 ( -7) =  40 + 140  =  180

So, we can observe that LHS = RHS

So, we can conclude that, the solution x = - 7 is correct.

There are some other examples are given below –

6

Example.1)  0.50 + -------- = 2

x

5

Ans.)  the given equation is 0.50 + -------- =  2

x

1              6

--------- + --------- =  2

2              x

x + 12

Or,  ------------- =  2

2x

By cross multiplication, we get  x + 12 =  4x

Or, 3x = 12

Or,  x = 4            (Ans.)

5                 6

Example.2) Solve  ----------- = -----------

x + 4            x – 4

5                6

Ans.) the given equation is  ---------- = ----------

x + 4           x – 4

By cross multiplication we get,  5 (x – 4) = 6 (x + 4)

Or,  5x – 20 = 6x + 24

Or,  6x – 5x = - 24 – 20

Or,  x  =  - 44               (Ans.)

2x + 5             3x - 5

Example.3)  Solve  ------------- = -------------

4x + 6             6x – 2

2x + 5            3x - 5

The given equation is, ------------ = ------------

4x + 6            6x – 2

By cross multiplication, (2x + 5)(6x – 2) = (3x – 5) (4x + 6)

Or, 12x² + 30x – 4x – 10 = 12x² - 20x + 18x – 30

Or,  26x – 10 = - 2x – 30

Or,  26x + 2x = 10 – 30

Or,  28x = - 20

Or,  x  = - 20/28  =  - 5/7          (Ans.)

1               2                6

Example.4) Solve ---------- + ---------- = -----------

x – 4           x + 5           2x – 5

1              2              6

Ans.) The given equation is, --------- + --------- = ----------

x – 4          x + 5         2x – 5

(x + 5) + 2 (x – 4)               6

---------------------- =  -----------

(x – 4) (x + 5)               2x – 5

x + 5 + 2x – 8               6

Or,  ------------------- = -------------

(x – 4) (x + 5)             2x – 5

(3x – 3)                   6

Or,  ------------------ = --------------

(x – 4) (x + 5)             2x – 5

By cross multiplication,

Or,  (3x – 3) (2x – 5)  =  6 (x – 4)(x + 5)

Or,  3x . 2x – 3 . 2x – 5 . 3x + 15  =  6 ( x² - 4x + 5x – 20)

Or,   6x² - 21x + 15 =  6 (x² + x – 20)

Or,   6x² - 21x + 15  = 6x² + 6x - 120

Or,       - 27x =  - 135

Or,           x  =  5                (Ans.)

x + 3            2x + 8

Example.5)   Solve  ------------ = ------------

2x – 5            4x – 5

x + 3              2x + 8

Ans.) The given equation is,  ------------ = --------------

2x – 5             4x – 5

By cross multiplication,

(x + 3)(4x – 5) =  (2x + 8)(2x – 5)

Or,   4x² + 12x – 5x – 15  =  4x² + 16x – 10x – 40

Or,    7x – 15 = 6x – 40

Or,    x  =  - 40 + 15 =  - 25            (Ans.)

3 (x+4)                  x + 5

Example.6)  Solve  4x - ------------ = 20 -  -----------

4                        20

3 (x+4)                  x + 5

The equation has been given, 4x - ------------- = 20 - ------------

4                        20

16x – 3 (x + 4)                (20 X 20) – (x + 5)

Or,  --------------------- =  ------------------------

4                                 20

By cross multiplication,

Or,   20 (16x – 3x – 12) =  4 (400 – x – 5)

Or,     20 (13x – 12) =  4 (395 – x)

Or,    260x – 240 = 1580 – 4x

Or,     256x  = 1580 + 240

Or,    256x =  1820

Or,    x =  7.1                   (Ans.)