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).

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

Add 35 to both sides.

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

Add 70 to both sides.

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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