Solving Linear Equations (D) Worksheet | Fun and Engaging PDF ... - Free Printable

Let’s solve each equation step by step. We’ll go section by section, and for each problem, we’ll isolate the variable (usually x) using basic algebra: adding, subtracting, multiplying, or dividing both sides equally.

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Section A

1)
(2x + 5)/3 = 11
Multiply both sides by 3:
2x + 5 = 33
Subtract 5:
2x = 28
Divide by 2:
x = 14

2)
(8 - 3x)/2 = 5
Multiply both sides by 2:
8 - 3x = 10
Subtract 8:
-3x = 2
Divide by -3:
x = -2/3

3)
(5 - 9x)/6 = -2
Multiply both sides by 6:
5 - 9x = -12
Subtract 5:
-9x = -17
Divide by -9:
x = 17/9

4)
(7x + 6)/3 - 9 = -12
Add 9 to both sides:
(7x + 6)/3 = -3
Multiply by 3:
7x + 6 = -9
Subtract 6:
7x = -15
Divide by 7:
x = -15/7

5)
8x + (1 - 4x)/8 = 7
Multiply every term by 8 to eliminate denominator:
64x + (1 - 4x) = 56
Simplify:
64x + 1 - 4x = 56 → 60x + 1 = 56
Subtract 1:
60x = 55
Divide by 60:
x = 55/60 = 11/12

6)
5/x = -6
Multiply both sides by x:
5 = -6x
Divide by -6:
x = -5/6

7)
11/(4x) + 9 = 3
Subtract 9:
11/(4x) = -6
Multiply both sides by 4x:
11 = -24x
Divide by -24:
x = -11/24

8)
5 - (3x)/4 = 8x
Multiply every term by 4:
20 - 3x = 32x
Add 3x:
20 = 35x
Divide by 35:
x = 20/35 = 4/7

9)
2 + (4x)/3 - 7 = 1
Combine constants:
(4x)/3 - 5 = 1
Add 5:
(4x)/3 = 6
Multiply by 3:
4x = 18
Divide by 4:
x = 18/4 = 9/2

10)
4 - (3x)/2 = 3x + 5
Multiply every term by 2:
8 - 3x = 6x + 10
Add 3x:
8 = 9x + 10
Subtract 10:
-2 = 9x
Divide by 9:
x = -2/9

11)
6 - 2/x = 10
Subtract 6:
-2/x = 4
Multiply both sides by x:
-2 = 4x
Divide by 4:
x = -2/4 = -1/2

12)
4 - (2x)/9 + x = -1
Combine like terms: write x as 9x/9
So: 4 + (-2x/9 + 9x/9) = -1 → 4 + (7x)/9 = -1
Subtract 4:
(7x)/9 = -5
Multiply by 9:
7x = -45
Divide by 7:
x = -45/7

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Section B

1)
4(2x - 3) = 8(2x + 5)
Expand both sides:
8x - 12 = 16x + 40
Subtract 8x:
-12 = 8x + 40
Subtract 40:
-52 = 8x
Divide by 8:
x = -52/8 = -13/2

2)
3(4x - 5) = 5(2x - 5)
Expand:
12x - 15 = 10x - 25
Subtract 10x:
2x - 15 = -25
Add 15:
2x = -10
Divide by 2:
x = -5

3)
8(6x + 2) = 5(x - 2)
Expand:
48x + 16 = 5x - 10
Subtract 5x:
43x + 16 = -10
Subtract 16:
43x = -26
Divide by 43:
x = -26/43

4)
2(3x - 4) = 7(11 - 2x)
Expand:
6x - 8 = 77 - 14x
Add 14x:
20x - 8 = 77
Add 8:
20x = 85
Divide by 20:
x = 85/20 = 17/4

5)
7(5 - x) = -4(x - 11)
Expand:
35 - 7x = -4x + 44
Add 7x:
35 = 3x + 44
Subtract 44:
-9 = 3x
Divide by 3:
x = -3

6)
-4(x - 8) = -6(4 + 3x)
Expand:
-4x + 32 = -24 - 18x
Add 18x:
14x + 32 = -24
Subtract 32:
14x = -56
Divide by 14:
x = -4

7)
7(4 - 3x) = 2(8x - 9) + 6
Expand left: 28 - 21x
Right: 16x - 18 + 6 = 16x - 12
So: 28 - 21x = 16x - 12
Add 21x:
28 = 37x - 12
Add 12:
40 = 37x
Divide by 37:
x = 40/37

8)
-6(3 - 4x) + 2x = 8(x + 11)
Left: -18 + 24x + 2x = -18 + 26x
Right: 8x + 88
So: -18 + 26x = 8x + 88
Subtract 8x:
-18 + 18x = 88
Add 18:
18x = 106
Divide by 18:
x = 106/18 = 53/9

9)
3(2x - 6) = 3 - 4(3 - x)
Left: 6x - 18
Right: 3 - 12 + 4x = -9 + 4x
So: 6x - 18 = 4x - 9
Subtract 4x:
2x - 18 = -9
Add 18:
2x = 9
Divide by 2:
x = 9/2

10)
9(2x - 1) - 3x = 3(12 + x)
Left: 18x - 9 - 3x = 15x - 9
Right: 36 + 3x
So: 15x - 9 = 36 + 3x
Subtract 3x:
12x - 9 = 36
Add 9:
12x = 45
Divide by 12:
x = 45/12 = 15/4

11)
4x - (2x - 8) = 5(1 + 2x)
Left: 4x - 2x + 8 = 2x + 8
Right: 5 + 10x
So: 2x + 8 = 5 + 10x
Subtract 2x:
8 = 5 + 8x
Subtract 5:
3 = 8x
Divide by 8:
x = 3/8

12)
10 - 6(8x - 2) = 9x - (3 + 4x)
Left: 10 - 48x + 12 = 22 - 48x
Right: 9x - 3 - 4x = 5x - 3
So: 22 - 48x = 5x - 3
Add 48x:
22 = 53x - 3
Add 3:
25 = 53x
Divide by 53:
x = 25/53

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Section C

1)
(5x - 2)/3 = (4x + 1)/2
Cross-multiply:
2(5x - 2) = 3(4x + 1)
10x - 4 = 12x + 3
Subtract 10x:
-4 = 2x + 3
Subtract 3:
-7 = 2x
Divide by 2:
x = -7/2

2)
(7x - 8)/5 = (2x + 5)/4
Cross-multiply:
4(7x - 8) = 5(2x + 5)
28x - 32 = 10x + 25
Subtract 10x:
18x - 32 = 25
Add 32:
18x = 57
Divide by 18:
x = 57/18 = 19/6

3)
(-8x - 1)/2 = (5 - 3x)/6
Cross-multiply:
6(-8x - 1) = 2(5 - 3x)
-48x - 6 = 10 - 6x
Add 48x:
-6 = 10 + 42x
Subtract 10:
-16 = 42x
Divide by 42:
x = -16/42 = -8/21

4)
[5(x + 11)]/3 = [3(1 + x)]/2
Cross-multiply:
2 * 5(x + 11) = 3 * 3(1 + x)
10(x + 11) = 9(1 + x)
10x + 110 = 9 + 9x
Subtract 9x:
x + 110 = 9
Subtract 110:
x = -101

5)
[3(2 + 5x)]/4 = [2(6x - 3)]/5
Cross-multiply:
5 * 3(2 + 5x) = 4 * 2(6x - 3)
15(2 + 5x) = 8(6x - 3)
30 + 75x = 48x - 24
Subtract 48x:
30 + 27x = -24
Subtract 30:
27x = -54
Divide by 27:
x = -2

6)
[2(3x - 5)]/3 = [-4(x - 2)]/7
Cross-multiply:
7 * 2(3x - 5) = 3 * (-4)(x - 2)
14(3x - 5) = -12(x - 2)
42x - 70 = -12x + 24
Add 12x:
54x - 70 = 24
Add 70:
54x = 94
Divide by 54:
x = 94/54 = 47/27

7)
(1/2)(2x - 6) = (1/4)(8 - 12x)
Multiply both sides by 4 to eliminate denominators:
2*(2x - 6) = 1*(8 - 12x)
4x - 12 = 8 - 12x
Add 12x:
16x - 12 = 8
Add 12:
16x = 20
Divide by 16:
x = 20/16 = 5/4

8)
(1/2)(5x + 7) = (3/4)(3x - 1)
Multiply both sides by 4:
2*(5x + 7) = 3*(3x - 1)
10x + 14 = 9x - 3
Subtract 9x:
x + 14 = -3
Subtract 14:
x = -17

9)
5/(3x + 1) = 12
Multiply both sides by (3x + 1):
5 = 12(3x + 1)
5 = 36x + 12
Subtract 12:
-7 = 36x
Divide by 36:
x = -7/36

10)
(x + 2)/(x + 3) = 4
Multiply both sides by (x + 3):
x + 2 = 4(x + 3)
x + 2 = 4x + 12
Subtract x:
2 = 3x + 12
Subtract 12:
-10 = 3x
Divide by 3:
x = -10/3

11)
(2x - 9)/(3x - 2) = -3
Multiply both sides by (3x - 2):
2x - 9 = -3(3x - 2)
2x - 9 = -9x + 6
Add 9x:
11x - 9 = 6
Add 9:
11x = 15
Divide by 11:
x = 15/11

12)
2/(3x + 10) = 1/(x - 1)
Cross-multiply:
2(x - 1) = 1(3x + 10)
2x - 2 = 3x + 10
Subtract 2x:
-2 = x + 10
Subtract 10:
x = -12

13)
2/(7x + 3) = 9/(2x - 5)
Cross-multiply:
2(2x - 5) = 9(7x + 3)
4x - 10 = 63x + 27
Subtract 4x:
-10 = 59x + 27
Subtract 27:
-37 = 59x
Divide by 59:
x = -37/59

14)
8/(6x + 12) = -11/(7x - 10)
Cross-multiply:
8(7x - 10) = -11(6x + 12)
56x - 80 = -66x - 132
Add 66x:
122x - 80 = -132
Add 80:
122x = -52
Divide by 122:
x = -52/122 = -26/61

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Final Answer:

Section A
1) 14
2) -2/3
3) 17/9
4) -15/7
5) 11/12
6) -5/6
7) -11/24
8) 4/7
9) 9/2
10) -2/9
11) -1/2
12) -45/7

Section B
1) -13/2
2) -5
3) -26/43
4) 17/4
5) -3
6) -4
7) 40/37
8) 53/9
9) 9/2
10) 15/4
11) 3/8
12) 25/53

Section C
1) -7/2
2) 19/6
3) -8/21
4) -101
5) -2
6) 47/27
7) 5/4
8) -17
9) -7/36
10) -10/3
11) 15/11
12) -12
13) -37/59
14) -26/61