LINEAR EQUATIONS IN ONE VARIABLE

Solve the equations. Write null or all where appropriate. Show fractions in simplest form.

Problem 1 :

5(y + 2) = 20

Solution :

5(y + 2) = 20

Use distributive property.

5y + 10 = 20

Subtract 10 from both sides.

5y + 10 – 10 = 20 – 10

5y = 10

Divide both sides by 5.

5y/5 = 10/5

y = 2

It has unique solution.

Problem 2 :

2(x – 4) = 2x – 8

Solution :

2(x – 4) = 2x – 8

Use distributive property.

2x – 8 = 2x – 8

Add 8 to both sides.

2x – 8 + 8 = 2x – 8 + 8

2x = 2x

Divide both sides by 2.

2x/2 = 2x/2

x = x

So, the solution is true for all numbers.

Problem 3 :

6(k + 3) = 2(4k + 5)

Solution :

6(k + 3) = 2(4k + 5)

Use distributive property on both sides.

6(k + 3) = 2(4k + 5)

6k + 18 = 8k + 10

Subtract 8k from both sides.

6k + 18 – 8k = 8k – 8k + 10

6k + 18 – 8k = 10

Subtract 18 from both sides.

6k + 18 – 8k - 18 = 10 -18

-2k = -8

Divide both sides by -2.

-2k/-2 = -8/-2

k = 4

So, it has one solution.

Problem 4 :

30 – 2x = 2(3x + 3)

Solution :

30 – 2x = 2(3x + 3)

Use distributive property.

30 – 2x = 6x + 6

Subtract 6x from both sides.

30 – 2x – 6x = 6x – 6x + 6

30 – 8x = 6

Subtract 30 from both sides.

30 – 8x - 30 = 6 – 30

-8x = -24

Divide both sides by -8.

-8x/-8 = -24/-8

x = 3

So, it has one solution.

Problem 5 :

16 = 4(n – 5) + 4

Solution :

16 = 4(n – 5) + 4

16 = 4n – 20 + 4

16 = 4n – 16

Add 16 to both sides.

16 + 16 = 4n – 16 + 16

32 = 4n

Divide both sides by 4.

32/4 = 4/4n

8 = n

So, it has one solution.

Problem 6 :

7p + 9 = 4(p – 3)

Solution :

7p + 9 = 4(p – 3)

Use distributive property.

7p + 9 = 4p – 12

Add 12 to both sides.

7p + 9 + 12 = 4p – 12 + 12

7p + 21 = 4p

Subtract 7p from both sides.

7p – 7p + 21 = 4p – 7p

21 = -3p

Divide both sides by -3.

-21/3 = -3p/-3

-7 = p

So, it has one solution.

Problem 7 :

3(2m + 4) = 6m + 15

Solution :

3(2m + 4) = 6m + 15

Use distributive property.

6m + 12 = 6m + 15

Subtract 6m from both sides.

6m – 6m + 12 = 6m – 6m + 15

12 = 15

So, it has null solution.

Problem 8 :

2(4w + 7) = 3(6 + 2w)

Solution :

2(4w + 7) = 3(6 + 2w)

Use distributive property on both sides.

8w + 14 = 18 + 6w

Subtract 14 from both sides.

8w + 14 - 14 = 18 + 6w – 14

8w = 4 + 6w

Subtract 6w from both sides.

8w – 6w = 4 + 6w – 6w

2w = 4

Divide both sides by 2.

2w/2 = 4/2

w = 2

So, it has one solution.

Problem 9 :

25 = 6(c + 7) - 14

Solution :

25 = 6(c + 7) - 14

25 = 6c + 42 – 14

25 = 6c + 28

Subtract 28 from both sides.

25 – 28 = 6c + 28 – 28

-3 = 6c

Divide both sides by 6.

-3/6 = 6c/6

-1/2 = c

So, it has one solution.

Problem 10 :

3(3h + h) = 2(h + 5)

Solution :

3(3h + h) = 2(h + 5)

Use distributive property on both sides.

9h + 3h = 2h + 10

12h = 2h + 10

Subtract 2h from both sides.

12h – 2h = 2h – 2h + 10

10h = 10

Divide both sides by 10.

10h/10 = 10/10

h = 1

So, it has one solution.

Problem 11 :

3(2 + n) = 10 + 2n

Solution :

3(2 + n) = 10 + 2n

Use distributive property.

6 + 3n = 10 + 2n

Subtract 6 from both sides.

6 + 3n - 6 = 10 + 2n – 6

3n = 4 + 2n

Subtract 2n from both sides.

3n - 2n = 4 + 2n – 2n

n = 4

So, it has one solution.