SOLVING RATIONAL EQUATIONS - SOLVING LINEAR EQUATIONS WITH FRACTIONAL AND DECIMAL COEFFICIENTS

Please study

Solving by Transposition before Solving Rational Equations

if you have not already done so.

There we discussed about Solving Linear Equations by Transposition.

That knowledge is a prerequisite here.

Linear Equations with fractional and decimal coefficients

The fractions should be cleared by multiplying both sides of the
equation by the Least Common Multiple (L.C.M.) of the
denominators of the fractions in the equation.

Let us see a few examples.

Example 1 of Solving Rational Equations

Solve the Equation x⁄2 + x⁄3 = 5.

Solution:
The given equation is x⁄2 + x⁄3 = 5.
L.C.M. of the denominators 2 and 3 is 6.
Multiplying both sides of the equation with 6, we get
6 x x⁄2 + 6 x x⁄3 = 6 x 5
⇒ 3x + 2x = 30 ⇒ 5x = 30 ⇒ x = 30⁄5 = 6. Ans.

Example 2 of Solving Rational Equations

Solve the Equation x⁄3 - x⁄9 = 7x⁄6 - 17⁄3

Solution:
The given equation is x⁄3 - x⁄9 = 7x⁄6 - 17⁄3
L.C.M. of the denominators 3, 6 and 9 is 18.
Multiplying both sides of the equation with 18, we get
18 x x⁄3 - 18 x x⁄9 = 18 x 7x⁄6 - 18 x 17⁄3
⇒ 6x - 2x = 3 x 7x - 6 x 17 ⇒ 4x = 21x - 102
⇒ 4x - 21x = -102 ⇒ -17x = -102 ⇒ x = 6. Ans.

In case of decimals, multiply both sides with 10^m
Where m = the maximum number of digits after the decimal place.

Example 3 of Solving Rational Equations

Solve the Equation 0.8x + 1.25 = 2x + 0.05

Solution:
The given equation is 0.8x + 1.25 = 2x + 0.05
Here maximum number of digits after the decimal place are 2.
∴ multiplying both sides with 10² i.e. 100, we get
100 (0.8x) + 100(1.25) = 100(2x) + 100(0.05)
⇒ 80x + 125 = 200x + 5 ⇒ 80x - 200x = 5 - 125
⇒ -120x = -120 ⇒ x = 1. Ans.

Example 4 of Solving Rational Equations

Solve the Equation 0.2(2x - 1) - 0.5(3x - 1) = 0.4

Solution:
The given equation is 0.2(2x - 1) - 0.5(3x - 1) = 0.4
Here maximum number of digits after the decimal place is 1.
∴ multiplying both sides with 10¹ i.e. 10, we get
10{ 0.2(2x - 1)} - 10{0.5(3x - 1)} = 10(0.4)
⇒ 2(2x - 1) - 5(3x - 1) = 4 ⇒ 4x - 2 - 15x + 5 = 4.
⇒ (4x - 15x) + (5 - 2) = 4 ⇒ -11x + 3 = 4
⇒ -11x = 4 - 3 = 1 ⇒ x = -1⁄11. Ans.

Exercise : Solving Rational Equations

1. Solve the Equation
   2x⁄3 - x⁄2 = 4
2. Solve the Equation
   (1.2)x + 0.04 = 2x - 3.56
3. Solve the Equation
   (3x - 2)⁄3 + (2x + 3)⁄2 = x + 7⁄6

Answers to Exercise : Solving Rational Equations

1. 24
2. 4.5
3. 1⁄3