TYPES OF SOLUTIONS LINEAR EQUATIONS IN ONE VARIABLE

Solve for x and find the types of solution for the equations given below. :

Problem 1 :

-5x+ 3 = 2x + 10

Solution :

-5x+ 3 = 2x + 10

Subtracting 2x on both sides

-5x - 2x + 3 = 10

-7x + 3 = 10

Subtracting 3 on both sides

-7x = 10 - 3

-7x = 7

Dividing by -7 on both sides.

x = -1

So, the value of x is -1 and it ha unique solution.

Problem 2 :

3x - 12x = 24 - 9x

Solution :

3x - 12x = 24 - 9x

-9x = 24 - 9x

Adding 9x on both sides

-9x + 9x = 24

0x = 24

No values of x will satisfy this equation. So, no solution.

Problem 3 :

4(2x - 3) + 4 = 8x - 8

Solution :

4(2x - 3) + 4 = 8x - 8

Using distributive property, we get

8x - 12 + 4 = 8x - 8

8x - 8 = 8x - 8

Subtracting 8x and add 8 on both sides

8x - 8x = -8 + 8

0x = 0

By observing the above step, it is very clear that all values of x will satisfy the equation. So, it has infinite number of solutions.

Problem 4 :

2(x + 7) = 6x + 9 - 4x

Solution :

2(x + 7) = 6x + 9 - 4x

Distributing 2, we get

2x + 14 = 6x - 4x + 9

2x + 14 = 2x + 9

Subtracting 2x and 14 on both sides

2x - 2x = 9 - 14

0x = -5

So, infinite number of solutions.

Problem 5 :

-5(3 - 4x) = -6 + 20x + 9

Solution :

-5(3 - 4x) = -6 + 20x + 9

Using distributive property, we get

-15 + 20x = -6 + 20x + 9

-15 + 20x = 3 + 20x

-15 - 3 = 20x - 20x

-18 = 0x

So, it has infinite number of solutions.

Problem 6 :

-8x + 3 - 2x = - 6x + 3 - 4x

Solution :

-8x + 3 - 2x = - 6x + 3 - 4x

-10x + 3 = -10x + 3

-10x + 10x = 3 - 3

0x = 0

So, it has infinite number of solutions.

Problem 7 :

10x + 3 + 10x = 13x - 3 + 7x

Solution :

10x + 3 + 10x = 13x - 3 + 7x

20x + 3 = 20x - 3

20x - 20x = -3 - 3

0x = -6

So, it has no solution.

Problem 8 :

0.125x = 0.025 (5x + 1)

Solution :

0.125x = 0.025 (5x + 1)

Using distributive property,

0.125x = 0.025(5x) + 0.025(1)

0.125x = 0.125x + 0.025

Subtracting 0.125x on both sides.

0.125x - 0.125x = 0.025

0x = 0.025

So, there is no solution.

Problem 9 :

The solution of which of the following equations is neither a fraction nor an integer.

(a) 3x + 2 = 5x + 2           (b) 4x – 18 = 2

(c) 4x + 7 = x + 2            (d) 5x – 8 = x + 4

Solution :

Option a :

3x + 2 = 5x + 2

3x - 5x = 2 - 2

0x = 0

has infinite number of solution.

Option b :

4x - 18 = 2

Adding 2 on both sides

4x = 2 + 18

4x = 20

Dividing by 4 on both sides

x = 20/4

x = 5

So, integer is the solution.

Option c :

4x + 7 = x + 2

4x - x = 2 - 7

3x = -5

x = -5/3

So, the solution is a neither nor a fraction. 

Option d :

5x - 8 = x + 4

5x - x = 4 + 8

4x = 12

x = 12/4

x = 3

So, the solution is a integer.

Problem 10 :

If 8x – 3 = 25 + 17x, then x is

(a) a fraction         (b) an integer

(c) a rational number      (d) cannot be solved

Solution :

8x – 3 = 25 + 17x

8x - 17x = 25 + 3

- 9x = 28

x = -28/9

So, the solution is a rational number.

Problem 11 :

The solution of the equation

3x – 4 = 1 – 2x is ____.

Solution :

3x – 4 = 1 – 2x

3x + 2x = 1 + 4

5x = 5

x = 5/5

x = 1

So, the value of x is 1.