SOLVE LINEAR EQUATIONS IN ONE VARIABLE

What is solution ?

The value which satisfies the given equation is known as solution. There are three types of solutions.

(i) Unique solution

(ii) Infinitely many solution

(iii)  No solution

Solve each equation. Indicate if an equation has no solution or infinitely many solutions.

Problem 4 :

b + 3b – 10 = 2(2b – 5)

Solution :

b + 3b – 10 = 2(2b – 5)

4b – 10 = 4b – 10

Add 10 to each side of the equation.

4b – 10 + 10 = 4b – 10 + 10

4b = 4b

So, it is infinitely many solutions.

Problem 5 :

y + 5y – 6 = 3(2y – 2)

Solution :

y + 5y – 6 = 3(2y – 2)

6y – 6 = 6y – 6

Add 6 to each side of the equation.

6y – 6 + 6 = 6y – 6 + 6

6y = 6y

So, it is infinitely many solutions.

Problem 6 :

-4i – 11 = 4i/2 + 7

Solution :

-4i – 11 = 4i/2 + 7

-4i – 11 = 2i + 7

Subtract 2i from each side of the equation.

-4i – 11 – 2i = 2i + 7 – 2i

-6i – 11 = 7

Add 11 to each side of the equation.

-6i – 11 + 11 = 7 + 11

-6i = 18

Divide each side by 6.

-6i/6 = 18/6

-i = 3

i = -3

So, it is one solution.

Problem 7 :

-4z + 6 = 4(-z + 2)

Solution :

-4z + 6 = 4(-z + 2)

-4z + 6 = -4z + 8

Add 4z to each side of the equation.

-4z + 6 + 4z = -4z + 8 + 4z

6 = 8

So, it is no solution.

Problem 8 :

4i + 3 = 4(i + 1)

Solution :

4i + 3 = 4(i + 1)

4i + 3 = 4i + 4

Subtract 4i from each side of the equation.

4i + 3 – 4i = 4i + 4 – 4i

3 = 4

So, it is no solution.

Problem 9 :

6z + z = 5z + 10

Solution :

6z + z = 5z + 10

7z = 5z + 10

Subtract 5z from each side of the equation.

7z – 5z = 5z – 5z + 10

2z = 10

Divide each side by 2.

2z/2 = 10/2

z = 5

So, it is one solution.

Problem 10 :

2c – c = 3c - 5

Solution :

2c – c = 3c - 5

c = 3c – 5

Subtract 3c from each side of the equation.

c – 3c = 3c – 3c – 5

-2c = -5

Divide each side by 2.

-2c/2 = -5/2

c = -5/2

So, it is one solution.

Problem 11 :

12n + 7 = 4n + 8n + 6

Solution :

12n + 7 = 4n + 8n + 6

12n + 7 = 12n + 6

Subtract 12n from each side of the equation.

12n + 7 – 12n = 12n + 6 – 12n

7 = 6

So, it is no solution.

Problem 12 :

12d(3 – 2) = 24

Solution :

12d(3 – 2) = 24

36d – 24d = 24

12d = 24

Divide each side by 12.

12d/12 = 24/12

d = 2

Problem 13 :

4x = x + 10.5

Solution :

4x = x + 10.5

Subtract x from each side of the equation.

4x - x = x - x + 10.5

3x = 10.5

Divide each side by 3.

3x/3 = 10.5/3

x = 3.5

So, it is one solution.