Solve the given linear equations on this printable worksheet.

Let’s solve each of the 8 linear equations step by step. The goal is to isolate the variable (usually `x`, or `q` in #8) on one side of the equation.

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①

Equation:
`5 + 3(x - 1) = 5x - 6`

Step 1: Distribute the 3
→ `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

Equation:
`5 - 3(5x + 2) = 4(7 - 3x) + 1`

Step 1: Distribute
Left: `5 - 15x - 6` → `-1 - 15x`
Right: `28 - 12x + 1` → `29 - 12x`

So:
→ `-1 - 15x = 29 - 12x`

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

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

Step 4: Divide by 3
→ `x = -10`

✔ Answer: x = -10

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③

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

This looks messy — let’s simplify term by term.

Step 1: Simplify each part

First term: `(3(7x - 1))/4 = (21x - 3)/4`

Second term: `(2x - (1 - x)/2)`
→ Write as: `2x - (1 - x)/2`
→ Get common denominator: `(4x)/2 - (1 - x)/2 = (4x - (1 - x))/2 = (4x - 1 + x)/2 = (5x - 1)/2`

So entire left side becomes:
`(21x - 3)/4 - (5x - 1)/2`

Step 2: Get common denominator (LCM of 4 and 2 is 4)

→ `(21x - 3)/4 - 2*(5x - 1)/4 = [21x - 3 - 10x + 2]/4 = (11x - 1)/4`

Now equation is:
`(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

→ `7x = 7`

Step 6: Divide by 7

→ `x = 1`

✔ Answer: x = 1

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④

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

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

So:
`9x² + 47x - 42 = 9x² + 18x + 8`

Step 3: Subtract 9x² from both sides

→ `47x - 42 = 18x + 8`

Step 4: Subtract 18x

→ `29x - 42 = 8`

Step 5: Add 42

→ `29x = 50`

Step 6: Divide by 29

→ `x = 50/29`

✔ Answer: x = 50/29

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⑤

Equation:
`(5x - 1)/2 - (x - 2)/6 = (2x + 4)/3`

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

Multiply every term 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

→ `10x - 1 = 8`

Step 3: Add 1

→ `10x = 9`

Step 4: Divide by 10

→ `x = 9/10`

✔ Answer: x = 9/10

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⑥

Equation:
`(3/4)x - 2 = (1/3)x + 3`

Step 1: Eliminate fractions — LCD of 4 and 3 is 12

Multiply every term by 12:

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

→ `9x - 24 = 4x + 36`

Step 2: Subtract 4x

→ `5x - 24 = 36`

Step 3: Add 24

→ `5x = 60`

Step 4: Divide by 5

→ `x = 12`

✔ Answer: x = 12

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⑦

Equation:
`0.12x + (0.5 + x)/2 = x/3 + 1.5`

Step 1: Convert decimals to fractions for easier work

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

So equation becomes:
`(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`

Now equation:
`(3/25)x + 1/4 + x/2 = x/3 + 3/2`

Step 2: Get all terms with x on left, constants on right

→ `(3/25)x + x/2 - x/3 = 3/2 - 1/4`

Step 3: Combine x terms — find LCD of 25, 2, 3 → 150

Convert each:

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

So left side:
`18/150 x + 75/150 x - 50/150 x = (18 + 75 - 50)/150 x = 43/150 x`

Right side:
`3/2 - 1/4 = 6/4 - 1/4 = 5/4`

So:
`43/150 x = 5/4`

Step 4: 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

*(You can leave as improper fraction or mixed number: 4 31/86)*

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⑧

Equation:
`0.5(3q + 87) = 1.5q + 43`

Step 1: Distribute 0.5

→ `1.5q + 43.5 = 1.5q + 43`

Step 2: Subtract 1.5q from both sides

→ `43.5 = 43`

✘ Contradiction!

This means no solution — the equation is inconsistent.

✔ Answer: No solution

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## ✔ Final Answers Summary:

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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