SOLVING LINEAR EQUATIONS WITH RATIONAL COEFFICIENTS

Problem 1 :

Solve:

x/2 + x/4 + x/5 + 10000 = x

Solution :

x/2 + x/4 + x/5 + 10000 = x

10000 = x/2 - x/4 - x/5

LCM of 2, 4, 5 is 20.

10000 = (20x-10x-5x-4x)/20

10000 = (20x-19x)/20

10000 = x/20

x/20 = 10000/1

Cross multiplying both sides.

x∙1 = 20∙10000

x = 200000

So, the answer is x = 200000.

Problem 2 :

If (5x/3) - 4 = 2x/5, then the numerical value of 2x-7 is

Solution :

(5x/3) - 4= 2x/5

Find common denominator

(5x - 12)/3= 2x/5

Doing cross multiplication.

5(5x-12) = 6x

25x-60 = 6x

25x-6x = 60

19x = 60

x = 60/19

The numerical value of 2x - 7= 2(60/19) - 7

= (120/19) - 7

= -13/19

Problem 3 :

-4/3y = -3/4 then y =

Solution:

Given, -4/3y = -3/4

y = (-3/4)×(-3/4)

y = (3/4)²

Hence, the value of y is (3/4)².

Problem 4 :

(x/5) + 30 = 18 has the solution as

Solution :

Transposing 30 to RHS and it becomes -30.

x/5 = 18-30

x/5 = -12

x = -12×5

x = -60

So, the solution as x = -60.

Problem 5 :

If (2/5) (x-2) = 5-(3/5)x then x = ?

Solution :

(2/5)(x-2)= 5 - (3/5)x

(2x-4) / 5 = (25-3x) / 5

Multiplying by 5 on both sides, we get

2x-4 = 25-3x

2x+3x = 25+4

5x = 29

x = 29/5

So, the answer is x = 29/5.

Problem 6 :

x/5 = (x - 1)/6

Solution :

Doing cross multiplication.

6x= 5(x-1)

6x= 5x-5

6x-5x = -5

x = -5

So, the answer is x= -5.

Problem 7:

(x/2) - (1/4) (x-1/3) = (1/6) (x+1) + (1/12)

Solution :

(x/2) - (1/4) (x-1/3) = (1/6) (x+1) + (1/12)

Combine like terms.

x/2 - x/4 + 1/12 = 1/6x + 1/6 + 1/12

LCM of 2, 4, 12 is 12 and LCM of 6, 6, 12 is 12.

(6x - 3x + 1)/12 = (2x + 2 + 1)/12

Simplify like terms.

(3x+1) / 12 = (2x+3) / 12

Multiplying by 12 on both sides, we get

3x + 1 = 2x + 3

3x - 2x = 3 - 1

x = 2

So, the answer is x = 2.

Problem 8 :

(1/2) (x + 1) + (1/3) (x - 1) = (5/12) (x - 2)

Solution :

Least common multiple of 2, 3, and 12 is 12.

[6(x + 1) + 4(x - 1)] / 12 = 5(x-2) / 12

Multiplying by 12 on both sides.

6(x + 1) + 4(x - 1) = 5(x-2)

Using distributive property.

6x + 6 + 4x - 4 = 5x - 10

10x + 2 = 5x - 10

5x = -10 - 2

x = -12/5