**Solve rational equations :**

A rational expression is a fraction with a polynomial in the numerator and denominator.If we have an equation containing rational expressions, we have a rational equation.

Solve rational equation means, we have to find the value of the variable in the given equation.

Let us see some example problems to understand the above concept.

**Example 1 :**

Solve

**Solution :**

To make the rational equation in a single line, we have make the denominators same. For that we have to take L.C.M.

x/6 = 9/20

Multiply 6 on both sides

x = (9/20) x 6

x = 54/20

Simplify using 2 times table

x = 27/10

Hence the solution is 27/10

**Example 2 :**

Solve

**Solution :**

To make the rational equation in a single line, we have make the denominators same. For that we have to take L.C.M.

L.C.M of (2, 4 and 6) = 12

Multiply both numerator and denominator the first fraction (n/2) by 6

Multiply both numerator and denominator the second fraction (3n/4) by 3

Multiply both numerator and denominator the third fraction (5n/6) by 2

(6n - 9n + 10n) / 12 = 21

7n/12 = 21

Multiply 12 on both sides

7 n = 21 x 12

Divide by 7 on both sides

n = (21 x 12)/7

n = 36

Hence the value of n is 36.

**Example 3 :**

Solve

**Solution :**

We have to consider x + 7 as (x + 7)/1

On the both sides in order to make the denominator same we have to take L.C.M

On the left side :

L.C.M of 1 and 3 = 3

[3(x + 7) - 8x]/3

On the right side :

L.C.M of 6 and 2 = 6

(17-15x)/6

[3(x + 7) - 8x]/3 = (17-15x)/6

(3x + 21 - 8x)/3 = (17-15x)/6

(-5x + 21)/3 = (17-15x)/6

Multiply by 6 on both sides

2(-5x + 21) = 17-15x

-10x + 42 = 17 - 15x

Add 15x on both sides

-10x + 15x + 42 = 17 - 15x + 15x

5x + 42 = 17

Subtract 42 on both sides

5x + 42 - 42 = 17 - 42

5x = -25

Divide by 5 on both sides

x = -25/5 ==> x = -5

Hence the value of x is -5.

**Example 4 :**

Solve

**Solution :**

Multiply by 3 on both sides

(x - 5) = 3(x - 3)/5

Multiply by 5 on both sides

5(x - 5) = 3(x - 3)

5x - 25 = 3x - 9

Subtract 3x on both sides

5x - 3x -25 = 3x - 3x - 9

2x - 25 = -9

Add 25 on both sides

2x - 25 + 25 = -9 + 25

2x = 16

Divide by 2 on both sides

x = 16/2 ==> x = 8

Hence the value of x is 8.

**Example 5 :**

Solve

**Solution :**

On the left side L.C.M of 4 and 3 is 12

[3(3t - 2) - 4(2t + 3)] / 12 ----(1)

On the right side, we have to consider t as t/1

L.C.M of 3 and 1 is 3

(2-3t)/3 ----(2)

(1) = (2)

(9t - 6 - 8t - 12)/12 = (2-3t)/3

(t - 18)/12 = (2-3t)/3

Multiply by 12 on both sides

t - 18 = [(2 - 3t)/3] x 12

t - 18 = 4(2 - 3t)

t - 18 = 8 - 12t

Add 12t on both sides

t - 18 + 12t = 8 - 12t + 12t

13t - 18 = 8

Add 18 on both sides

13t - 18 + 18 = 8 + 18

13t = 26

Divide by 13 on both sides

t = 26/13 ==> t = 2

Hence the value of t is 2.

After having gone through the stuff given above, we hope that the students would have understood "Solve rational equations".

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