Linear Equations Worksheets with Answer Key - Free Printable

Let's solve each linear equation step by step.

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① 5 + 3(x - 1) = 5x - 6

Step 1: Distribute the 3 on the left.
→ 5 + 3x - 3 = 5x - 6

Step 2: Combine like terms on left.
→ 2 + 3x = 5x - 6

Step 3: Subtract 3x from both sides.
→ 2 = 2x - 6

Step 4: Add 6 to both sides.
→ 8 = 2x

Step 5: Divide by 2.
→ x = 4

✔ Answer: x = 4

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② 5 - 3(5x + 2) = 4(7 - 3x) + 1

Step 1: Distribute both sides.
Left: 5 - 15x - 6 = -15x -1
Right: 28 - 12x + 1 = -12x + 29

So:
→ -15x - 1 = -12x + 29

Step 2: Add 15x to both sides.
→ -1 = 3x + 29

Step 3: Subtract 29 from both sides.
→ -30 = 3x

Step 4: Divide by 3.
→ x = -10

✔ Answer: x = -10

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③ (3(7x - 1))/4 - (2x - (1 - x)/2) = x + 3/2

This one is messy — let’s simplify step by step.

First, write it clearly:

> [3(7x - 1)] / 4 - [2x - (1 - x)/2] = x + 3/2

Step 1: Simplify inside parentheses and distribute.

Left term: [21x - 3]/4

Second term: 2x - (1 - x)/2 → Let’s combine this part.

Write 2x as (4x)/2 to have common denominator:

→ (4x)/2 - (1 - x)/2 = [4x - (1 - x)] / 2 = [4x -1 + x]/2 = (5x - 1)/2

So now the equation becomes:

> (21x - 3)/4 - (5x - 1)/2 = x + 3/2

Step 2: Get a common denominator for left side. LCD of 4 and 2 is 4.

→ (21x - 3)/4 - 2*(5x - 1)/4 = x + 3/2

→ [21x - 3 - 10x + 2]/4 = x + 3/2

→ (11x - 1)/4 = x + 3/2

Step 3: Multiply both sides by 4 to eliminate denominator.

→ 11x - 1 = 4x + 6

Step 4: Subtract 4x from both sides.

→ 7x - 1 = 6

Step 5: Add 1 to both sides.

→ 7x = 7

→ x = 1

✔ Answer: x = 1

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④ (9x - 7)/(3x + 4) = (3x + 2)/(x + 6)

This is a rational equation. Cross-multiply.

Step 1: Cross multiply.

→ (9x - 7)(x + 6) = (3x + 2)(3x + 4)

Step 2: Expand both sides.

Left:
9x(x + 6) -7(x + 6) = 9x² + 54x -7x -42 = 9x² + 47x - 42

Right:
3x(3x + 4) + 2(3x + 4) = 9x² + 12x + 6x + 8 = 9x² + 18x + 8

Now set equal:

→ 9x² + 47x - 42 = 9x² + 18x + 8

Step 3: Subtract 9x² from both sides.

→ 47x - 42 = 18x + 8

Step 4: Subtract 18x from both sides.

→ 29x - 42 = 8

Step 5: Add 42 to both sides.

→ 29x = 50

→ x = 50/29

✔ Answer: x = 50/29

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⑤ (5x - 1)/2 - (x - 2)/6 = (2x + 4)/3

Step 1: Find LCD of denominators 2, 6, 3 → LCD = 6

Multiply entire equation by 6:

→ 6 * [(5x - 1)/2] - 6 * [(x - 2)/6] = 6 * [(2x + 4)/3]

Simplify:

→ 3(5x - 1) - (x - 2) = 2(2x + 4)

→ 15x - 3 - x + 2 = 4x + 8

→ 14x -1 = 4x + 8

Step 2: Subtract 4x from both sides.

→ 10x -1 = 8

Step 3: Add 1.

→ 10x = 9

→ x = 9/10

✔ Answer: x = 9/10

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⑥ (3/4)x - 2 = (1/3)x + 3

Step 1: Eliminate fractions by multiplying entire equation by LCD of 4 and 3 → 12.

→ 12 * [(3/4)x - 2] = 12 * [(1/3)x + 3]

→ 9x - 24 = 4x + 36

Step 2: Subtract 4x from both sides.

→ 5x - 24 = 36

Step 3: Add 24.

→ 5x = 60

→ x = 12

✔ Answer: x = 12

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⑦ 0.12x + (0.5 + x)/2 = x/3 + 1.5

Step 1: Eliminate decimals. Multiply entire equation by 100 to make everything integer.

But better: convert decimals to fractions.

0.12 = 12/100 = 3/25
0.5 = 1/2
1.5 = 3/2

So:

→ (3/25)x + (1/2 + x)/2 = x/3 + 3/2

Simplify middle term: (1/2 + x)/2 = (1/2)/2 + x/2 = 1/4 + x/2

Equation becomes:

→ (3/25)x + 1/4 + x/2 = x/3 + 3/2

Step 2: Move all terms to left side.

→ (3/25)x + x/2 - x/3 + 1/4 - 3/2 = 0

Combine x terms and constants.

First, constants: 1/4 - 3/2 = 1/4 - 6/4 = -5/4

Now x terms: find LCD of 25, 2, 3 → 150

Convert:

→ (3/25)x = (18/150)x
→ (1/2)x = (75/150)x
→ -(1/3)x = -(50/150)x

Add: 18 + 75 - 50 = 43 → (43/150)x

So:

→ (43/150)x - 5/4 = 0

→ (43/150)x = 5/4

Step 3: Solve for x.

Multiply both sides by 150:

→ 43x = (5/4)*150 = (5*150)/4 = 750/4 = 375/2

→ x = (375/2) / 43 = 375/(2*43) = 375/86

✔ Answer: x = 375/86

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⑧ 0.5(3q + 87) = 1.5q + 43

Step 1: Distribute 0.5 on left.

→ 1.5q + 43.5 = 1.5q + 43

Step 2: Subtract 1.5q from both sides.

→ 43.5 = 43

This is a contradiction!

✔ Answer: No solution

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## ✔ Final Answers:

① x = 4
② x = -10
③ x = 1
④ x = 50/29
⑤ x = 9/10
⑥ x = 12
⑦ x = 375/86
⑧ No solution

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