# SOLVING EQUATIONS WITH RATIONAL NUMBERS

In this section, we will learn how to solve an equation with variables on both sides that involves rationals numbers (fractions and terminating decimals).

When we have fraction in an equation, our first aim is to get of the fraction. Once we get rid of the fraction, the equation can be solved easily.

Case (i) :

If you have one fraction, multiply both sides of the equation by the denominator to get rid of the fraction.

Case (ii) :

If you have more than one fraction with same denominator, multiply both sides of the equation by the denominator to get rid of the fractions.

Case (iii) :

If you have more than one fraction with different denominators, find the least common multiple of the denominators, then multiply both sides of the equation by the least common multiple of the denominators to get rid of the fractions.

Example 1 :

Solve for n :

Solution :

In the equation above, there is only one fraction with the denominator 5.

So, multiply both sides of the equation by 3 to get rid of the fraction.

2n = 20

Divide both sides by 2.

n = 10

Example 2 :

Solve for x :

Solution :

In the equation above, there is only one fraction with the denominator 3.

So, multiply both sides of the equation by 3 to get rid of the fraction.

Use the distributive property on the left side and solve for x.

5x + 3 = 3x + 9

Subtract 3x from both sides.

2x + 3 = 9

Subtract 3 from both sides.

2x = 6

Divide both sides by 2.

x = 3

Example 3 :

Solve for y :

Solution :

Multiply both sides of the equation by 7 to get rid of the fractions.

y - 42 = 3y + 28

Subtract y from both sides.

-42 = 2y + 28

Subtract 28 from both sides.

-70 = 2y

Divide both sides by 2.

-35 = y

Example 4 :

Solve for x :

Solution :

Least common multiple of (10, 2, 5) = 10.

Multiply both sides of the equation by 10 to get rid of the fractions.

Subtract 6x from both sides.

x + 15 = 20

Subtract 15 from both sides.

x = 5

Example 5 :

Solve for k :

Solution :

Least common multiple of (5, 4) = 20.

Multiply both sides of the equation by 20 to get rid of the fractions.

16k - 35 = 4k + 5k

16k - 35 = 9k

Subtract 9k from both sides.

7k - 35 = 0

7k = 35

Divide both sides by 7.

k = 5

Example 6 :

Solve for m :

Solution :

Least common multiple of (7, 5) = 35.

Multiply both sides of the equation by 35 to get rid of the fractions.

5m - 20 = 7m - 70

Subtract 5m from both sides.

-20 = 2m - 70

50 = 2m

Divide both sides by 2.

25 = m

Example 7 :

Solve for z :

Solution :

Least common multiple of (2, 9) = 18.

Multiply both sides of the equation by 18 to get rid of the fractions.

9(z - 2) + 2(z + 10) = 90

9z - 18 + 2z + 20 = 90

11z + 2 = 90

Subtract 2 from both sides.

11z = 88

Divide both sides by 11.

z = 8

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