Linear Equations Worksheets with Answer Key - Free Printable

Let’s solve each equation one by one, step by step. We’ll simplify both sides, move variables to one side and numbers to the other, then solve for the variable.

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**Problem 1:
5 + 3(x - 1) = 5x - 6**

Step 1: Distribute the 3 on the left side.
5 + 3x - 3 = 5x - 6
→ (5 - 3) + 3x = 5x - 6
→ 2 + 3x = 5x - 6

Step 2: Move all x terms to one side. Subtract 3x from both sides.
2 = 2x - 6

Step 3: Move constants to the other side. Add 6 to both sides.
8 = 2x

Step 4: Divide by 2.
x = 4

✔ Check: Left side: 5 + 3(4-1) = 5 + 9 = 14
Right side: 5*4 - 6 = 20 - 6 = 14 → Correct!

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**Problem 2:
5 - 3(5x + 2) = 4(7 - 3x) + 1**

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

So:
-15x - 1 = 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

✔ Check: Left: 5 - 3(5*(-10)+2) = 5 - 3(-50+2) = 5 - 3(-48) = 5 + 144 = 149
Right: 4(7 - 3*(-10)) + 1 = 4(7+30) + 1 = 4*37 + 1 = 148 + 1 = 149 → Correct!

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**Problem 3:
[3(7x - 1)]/4 - (2x - (1 - x)/2) = x + 3/2**

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

First, write it clearly:

\[\frac{3(7x - 1)}{4} - \left(2x - \frac{1 - x}{2}\right) = x + \frac{3}{2}\]

Step 1: Simplify inside the parentheses.

Inside the big parenthesis:
2x - (1 - x)/2 → Let’s get common denominator.

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

So now the equation is:

\[\frac{3(7x - 1)}{4} - \frac{5x - 1}{2} = x + \frac{3}{2}\]

Step 2: Get rid of denominators. Multiply EVERY term by 4 (LCM of 4 and 2).

4 * [3(7x-1)/4] = 3(7x - 1)
4 * [-(5x - 1)/2] = -2*(5x - 1)
4 * x = 4x
4 * (3/2) = 6

So:

3(7x - 1) - 2(5x - 1) = 4x + 6

Step 3: Expand.

21x - 3 - 10x + 2 = 4x + 6
→ (21x - 10x) + (-3 + 2) = 4x + 6
→ 11x - 1 = 4x + 6

Step 4: Subtract 4x from both sides.
7x - 1 = 6

Step 5: Add 1 to both sides.
7x = 7

Step 6: Divide by 7.
x = 1

✔ Check: Plug x=1 into original.

Left: [3(7*1 -1)]/4 - (2*1 - (1-1)/2) = [3*6]/4 - (2 - 0) = 18/4 - 2 = 4.5 - 2 = 2.5
Right: 1 + 3/2 = 2.5 → Correct!

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

This is a proportion — cross multiply.

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

Step 1: 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 2: Subtract 9x² from both sides.

47x - 42 = 18x + 8

Step 3: Subtract 18x from both sides.

29x - 42 = 8

Step 4: Add 42 to both sides.

29x = 50

Step 5: Divide by 29.

x = 50/29

✔ Check: Not needed unless asked, but we can verify if denominators are not zero.

At x = 50/29 ≈ 1.72,
3x+4 ≠ 0, x+6 ≠ 0 → OK.

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**Problem 5:
(5x - 1)/2 - (x - 2)/6 = (2x + 4)/3**

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

Multiply every term by 6:

6*(5x-1)/2 = 3*(5x-1) = 15x - 3
6*(x-2)/6 = 1*(x-2) = x - 2 → but note: it's MINUS this term → so - (x - 2)
6*(2x+4)/3 = 2*(2x+4) = 4x + 8

So equation becomes:

15x - 3 - (x - 2) = 4x + 8

Step 2: Distribute the negative sign.

15x - 3 - x + 2 = 4x + 8
→ 14x - 1 = 4x + 8

Step 3: Subtract 4x from both sides.

10x - 1 = 8

Step 4: Add 1 to both sides.

10x = 9

Step 5: Divide by 10.

x = 9/10 or 0.9

✔ Check: Plug x = 0.9

Left: (5*0.9 -1)/2 - (0.9 - 2)/6 = (4.5 -1)/2 - (-1.1)/6 = 3.5/2 + 1.1/6 = 1.75 + 0.1833... ≈ 1.9333
Right: (2*0.9 + 4)/3 = (1.8 + 4)/3 = 5.8/3 ≈ 1.9333 → Correct!

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**Problem 6:
(3/4)x - 2 = (1/3)x + 3**

Step 1: Eliminate fractions. LCM of 4 and 3 is 12.

Multiply every term by 12:

12*(3/4 x) = 9x
12*(-2) = -24
12*(1/3 x) = 4x
12*3 = 36

Equation:
9x - 24 = 4x + 36

Step 2: Subtract 4x from both sides.

5x - 24 = 36

Step 3: Add 24 to both sides.

5x = 60

Step 4: Divide by 5.

x = 12

✔ Check: Left: (3/4)*12 - 2 = 9 - 2 = 7
Right: (1/3)*12 + 3 = 4 + 3 = 7 → Correct!

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**Problem 7:
0.12x + (0.5 + x)/2 = x/3 + 1.5**

Step 1: Eliminate decimals and fractions. Multiply everything by 6 (LCM of 2 and 3).

6 * 0.12x = 0.72x
6 * [(0.5 + x)/2] = 3*(0.5 + x) = 1.5 + 3x
6 * (x/3) = 2x
6 * 1.5 = 9

Equation becomes:

0.72x + 1.5 + 3x = 2x + 9

Step 2: Combine like terms on left.

(0.72x + 3x) + 1.5 = 2x + 9
→ 3.72x + 1.5 = 2x + 9

Step 3: Subtract 2x from both sides.

1.72x + 1.5 = 9

Step 4: Subtract 1.5 from both sides.

1.72x = 7.5

Step 5: Divide by 1.72.

x = 7.5 / 1.72

Let’s compute that:

7.5 ÷ 1.72 = ?

Multiply numerator and denominator by 100 to eliminate decimals:

750 / 172

Simplify fraction:

Divide numerator and denominator by 2: 375 / 86

That’s about 4.360… but let’s keep as exact fraction.

Wait — maybe better to use fractions from start?

Original: 0.12x = 12/100 x = 3/25 x
0.5 = 1/2
1.5 = 3/2

Rewrite equation:

(3/25)x + (1/2 + x)/2 = x/3 + 3/2

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

So:

(3/25)x + 1/4 + x/2 = x/3 + 3/2

Now combine x terms on left:

x*(3/25 + 1/2) + 1/4 = x/3 + 3/2

Compute 3/25 + 1/2 = (6 + 25)/50 = 31/50

So:

(31/50)x + 1/4 = (1/3)x + 3/2

Move all x to left, constants to right.

(31/50)x - (1/3)x = 3/2 - 1/4

Compute left: LCM of 50 and 3 is 150.

(93/150 - 50/150)x = 43/150 x

Right: 6/4 - 1/4 = 5/4

So:

43/150 x = 5/4

Multiply both sides by 150:

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

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

Same as before.

375 ÷ 86 = 4.360... but let’s leave as fraction.

But wait — earlier decimal method gave same result.

Actually, 7.5 / 1.72 = 750 / 172 = 375 / 86 → yes.

So x = 375/86

We can check if reducible? 375 and 86.

86 = 2*43
375 = 3*5^3 → no common factors → so x = 375/86

But perhaps they want decimal? Or did I make mistake?

Wait — let me recheck with decimals.

From earlier:

After multiplying by 6:
0.72x + 1.5 + 3x = 2x + 9
→ 3.72x + 1.5 = 2x + 9
→ 1.72x = 7.5
→ x = 7.5 / 1.72

Calculate: 1.72 * 4 = 6.88
7.5 - 6.88 = 0.62
1.72 * 0.36 = approx 0.6192 → so x ≈ 4.36

But 375/86 = 4.360465... → correct.

However, let’s see if problem expects simplified fraction or decimal.

Since others were integers or simple fractions, maybe I made error in setup.

Alternative approach:

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

Write all as 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

As above.

(1/2 + x)/2 = 1/4 + x/2

So:

(3/25)x + x/2 + 1/4 = x/3 + 3/2

Combine x terms: x(3/25 + 1/2) = x(6/50 + 25/50) = x(31/50)

So:

(31/50)x + 1/4 = (1/3)x + 3/2

Subtract (1/3)x and 1/4 from both sides:

(31/50 - 1/3)x = 3/2 - 1/4

Compute:

31/50 - 1/3 = (93 - 50)/150 = 43/150

3/2 - 1/4 = 6/4 - 1/4 = 5/4

So:

(43/150)x = 5/4

x = (5/4) * (150/43) = (5 * 150) / (4 * 43) = 750 / 172 = 375 / 86

Yes, correct.

So x = 375/86

But let’s reduce? 375 and 86 share no common factors → so final answer is 375/86.

Alternatively, mixed number: 4 and 31/86, but improper fraction is fine.

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**Problem 8:
0.5(3q + 87) = 1.5q + 43**

Step 1: Distribute 0.5 on left.

0.5 * 3q = 1.5q
0.5 * 87 = 43.5

So:
1.5q + 43.5 = 1.5q + 43

Step 2: Subtract 1.5q from both sides.

43.5 = 43

This is FALSE. Contradiction.

That means NO SOLUTION.

Wait — let me double-check.

0.5 * 87 = 43.5, yes.
Right side: 1.5q + 43

So after subtracting 1.5q: 43.5 = 43 → never true.

So no value of q satisfies this.

But let’s check if typo? Maybe 0.5(3q + 86)? But as written, it’s 87.

Perhaps intended to be equal? But mathematically, it’s inconsistent.

So answer: No solution.

But let’s confirm.

Suppose q = anything, say q=0:
Left: 0.5*(0 + 87) = 43.5
Right: 0 + 43 = 43 → not equal.

q=10: Left: 0.5*(30+87)=0.5*117=58.5
Right: 1.5*10 +43=15+43=58 → still off by 0.5.

Always differs by 0.5 → so indeed, no solution.

Final Answer for #8: No solution

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Now compiling all 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

Final Answer:
1. 4
2. -10
3. 1
4. 50/29
5. 9/10
6. 12
7. 375/86
8. No solution