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.)