Linear Equations Worksheets with Answer Key - Free Printable

I'll solve each linear equation step by step, working through them carefully.

Problem 1: 8(4a - 4) - 1 = 14 - 8a

Step 1: Distribute the 8 on the left side
32a - 32 - 1 = 14 - 8a

Step 2: Combine like terms on the left
32a - 33 = 14 - 8a

Step 3: Add 8a to both sides
40a - 33 = 14

Step 4: Add 33 to both sides
40a = 47

Step 5: Divide by 40
a = 47/40 or a = 1.175

Let me verify: 8(4 × 1.175 - 4) - 1 = 8(4.7 - 4) - 1 = 8(0.7) - 1 = 5.6 - 1 = 4.6
Right side: 14 - 8(1.175) = 14 - 9.4 = 4.6 ✓

Problem 2: 4(-8x + 5) = -30x - 26

Step 1: Distribute the 4 on the left
-32x + 20 = -30x - 26

Step 2: Add 32x to both sides
20 = 2x - 26

Step 3: Add 26 to both sides
46 = 2x

Step 4: Divide by 2
x = 23

Let me verify: Left: 4(-8 × 23 + 5) = 4(-184 + 5) = 4(-179) = -716
Right: -30(23) - 26 = -690 - 26 = -716 ✓

Problem 3: 27 = 46 + 3x - x

Step 1: Combine like terms on the right
27 = 46 + 2x

Step 2: Subtract 46 from both sides
-19 = 2x

Step 3: Divide by 2
x = -19/2 or x = -9.5

Let me verify: Right side: 46 + 3(-9.5) - (-9.5) = 46 - 28.5 + 9.5 = 27 ✓

Problem 4: 38 + 7k = 8(k + 4)

Step 1: Distribute the 8 on the right
38 + 7k = 8k + 32

Step 2: Subtract 7k from both sides
38 = k + 32

Step 3: Subtract 32 from both sides
6 = k

So k = 6

Let me verify: Left: 38 + 7(6) = 38 + 42 = 80
Right: 8(6 + 4) = 8(10) = 80 ✓

Problem 5: -2(4 - x) = 6(x + 4) + 4x

Step 1: Distribute on both sides
-8 + 2x = 6x + 24 + 4x

Step 2: Combine like terms on the right
-8 + 2x = 10x + 24

Step 3: Subtract 2x from both sides
-8 = 8x + 24

Step 4: Subtract 24 from both sides
-32 = 8x

Step 5: Divide by 8
x = -4

Let me verify: Left: -2(4 - (-4)) = -2(4 + 4) = -2(8) = -16
Right: 6(-4 + 4) + 4(-4) = 6(0) - 16 = -16 ✓

Problem 6: 0.05z + 0.2 = 0.15z - 10.5

Step 1: Subtract 0.05z from both sides
0.2 = 0.1z - 10.5

Step 2: Add 10.5 to both sides
10.7 = 0.1z

Step 3: Divide by 0.1 (or multiply by 10)
z = 107

Let me verify: Left: 0.05(107) + 0.2 = 5.35 + 0.2 = 5.55
Right: 0.15(107) - 10.5 = 16.05 - 10.5 = 5.55 ✓

Problem 7: 4(6 + 4x) = 0

Step 1: Distribute the 4
24 + 16x = 0

Step 2: Subtract 24 from both sides
16x = -24

Step 3: Divide by 16
x = -24/16 = -3/2 or x = -1.5

Let me verify: 4(6 + 4(-1.5)) = 4(6 - 6) = 4(0) = 0 ✓

Problem 8: x/8 - x/9 = 1

Step 1: Find common denominator (72)
9x/72 - 8x/72 = 1

Step 2: Combine fractions
x/72 = 1

Step 3: Multiply both sides by 72
x = 72

Let me verify: 72/8 - 72/9 = 9 - 8 = 1 ✓

Final Answer:
① a = 47/40 (or 1.175)
② x = 23
③ x = -19/2 (or -9.5)
④ k = 6
⑤ x = -4
⑥ z = 107
⑦ x = -3/2 (or -1.5)
⑧ x = 72