Linear Equations Worksheet featuring eight problems to solve, designed for math practice.

Let's solve each of the linear equations step by step.

---

Problem 1:

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

Step 1: Expand the left side:
$$
5 + 3x - 3 = 5x - 6
$$
$$
3x + 2 = 5x - 6
$$

Step 2: Move all $ x $-terms to one side and constants to the other:
Subtract $ 3x $ from both sides:
$$
2 = 2x - 6
$$

Add 6 to both sides:
$$
8 = 2x
$$

Divide by 2:
$$
x = 4
$$

✔ Answer: $ x = 4 $

---

Problem 2:

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

Step 1: Expand both sides:
Left:
$$
5 - 15x - 6 = -15x - 1
$$

Right:
$$
28 - 12x + 1 = 29 - 12x
$$

So equation becomes:
$$
-15x - 1 = 29 - 12x
$$

Step 2: Move variables to one side:
Add $ 15x $ to both sides:
$$
-1 = 29 + 3x
$$

Subtract 29:
$$
-30 = 3x
$$

Divide by 3:
$$
x = -10
$$

✔ Answer: $ x = -10 $

---

Problem 3:

Equation:
$$
\frac{3(7x - 1)}{4} - \left(2x - \frac{1 - x}{2}\right) = x + \frac{3}{2}
$$

Step 1: Simplify each part carefully.

First, simplify $ \frac{3(7x - 1)}{4} $:
$$
= \frac{21x - 3}{4}
$$

Now simplify $ 2x - \frac{1 - x}{2} $:
$$
= 2x - \frac{1}{2} + \frac{x}{2} = \left(2x + \frac{x}{2}\right) - \frac{1}{2} = \frac{5x}{2} - \frac{1}{2}
$$

So the full equation is:
$$
\frac{21x - 3}{4} - \left(\frac{5x}{2} - \frac{1}{2}\right) = x + \frac{3}{2}
$$

Distribute the minus sign:
$$
\frac{21x - 3}{4} - \frac{5x}{2} + \frac{1}{2} = x + \frac{3}{2}
$$

Step 2: Eliminate denominators. Multiply every term by 4 (LCM of 4, 2):
$$
4 \cdot \left(\frac{21x - 3}{4}\right) - 4 \cdot \left(\frac{5x}{2}\right) + 4 \cdot \left(\frac{1}{2}\right) = 4 \cdot x + 4 \cdot \frac{3}{2}
$$

$$
(21x - 3) - 10x + 2 = 4x + 6
$$

Simplify left side:
$$
(21x - 10x) + (-3 + 2) = 11x - 1
$$

So:
$$
11x - 1 = 4x + 6
$$

Step 3: Solve:
Subtract $ 4x $:
$$
7x - 1 = 6
$$

Add 1:
$$
7x = 7
$$

Divide:
$$
x = 1
$$

✔ Answer: $ x = 1 $

---

Problem 4:

Equation:
$$
\frac{9x - 7}{3x + 4} = \frac{3x + 2}{x + 6}
$$

This is a rational equation. Cross-multiply:

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

Step 1: Expand both sides:

Left:
$$
9x(x) + 9x(6) -7(x) -7(6) = 9x^2 + 54x - 7x - 42 = 9x^2 + 47x - 42
$$

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

Now set equal:
$$
9x^2 + 47x - 42 = 9x^2 + 18x + 8
$$

Subtract $ 9x^2 $ from both sides:
$$
47x - 42 = 18x + 8
$$

Subtract $ 18x $:
$$
29x - 42 = 8
$$

Add 42:
$$
29x = 50
$$

Divide:
$$
x = \frac{50}{29}
$$

Check for restrictions:
Denominators: $ 3x + 4 \neq 0 $ → $ x \neq -\frac{4}{3} $
$ x + 6 \neq 0 $ → $ x \neq -6 $
$ \frac{50}{29} $ is fine.

✔ Answer: $ x = \frac{50}{29} $

---

Problem 5:

Equation:
$$
\frac{5x - 1}{2} - \frac{x - 2}{6} = \frac{2x + 4}{3}
$$

Step 1: Find LCM of denominators: 6

Multiply entire equation by 6:
$$
6 \cdot \left(\frac{5x - 1}{2}\right) - 6 \cdot \left(\frac{x - 2}{6}\right) = 6 \cdot \left(\frac{2x + 4}{3}\right)
$$

$$
3(5x - 1) - (x - 2) = 2(2x + 4)
$$

Expand:
$$
15x - 3 - x + 2 = 4x + 8
$$

Simplify left:
$$
14x - 1 = 4x + 8
$$

Subtract $ 4x $:
$$
10x - 1 = 8
$$

Add 1:
$$
10x = 9
$$

Divide:
$$
x = \frac{9}{10}
$$

✔ Answer: $ x = \frac{9}{10} $

---

Problem 6:

Equation:
$$
\frac{3}{4}x - 2 = \frac{1}{3}x + 3
$$

Step 1: Eliminate fractions. LCM of 4 and 3 is 12. Multiply all terms by 12:

$$
12 \cdot \left(\frac{3}{4}x\right) - 12 \cdot 2 = 12 \cdot \left(\frac{1}{3}x\right) + 12 \cdot 3
$$

$$
9x - 24 = 4x + 36
$$

Step 2: Solve:
Subtract $ 4x $:
$$
5x - 24 = 36
$$

Add 24:
$$
5x = 60
$$

Divide:
$$
x = 12
$$

✔ Answer: $ x = 12 $

---

Problem 7:

Equation:
$$
0.12x + \frac{0.5 + x}{2} = \frac{x}{3} + 1.5
$$

Step 1: Simplify each term.

Left:
$$
0.12x + \frac{0.5}{2} + \frac{x}{2} = 0.12x + 0.25 + 0.5x = 0.62x + 0.25
$$

Right:
$$
\frac{x}{3} + 1.5
$$

So:
$$
0.62x + 0.25 = \frac{x}{3} + 1.5
$$

Step 2: Eliminate decimals. Multiply everything by 100 to eliminate decimals:
$$
62x + 25 = \frac{100x}{3} + 150
$$

Now multiply entire equation by 3 to eliminate fraction:
$$
3(62x + 25) = 100x + 450
$$

$$
186x + 75 = 100x + 450
$$

Subtract $ 100x $:
$$
86x + 75 = 450
$$

Subtract 75:
$$
86x = 375
$$

Divide:
$$
x = \frac{375}{86} \approx 4.36
$$

But let's keep as fraction:
$$
x = \frac{375}{86}
$$

Check if reducible: GCD of 375 and 86?
86 = 2×43; 375 = 3×5³ → no common factors.

✔ Answer: $ x = \frac{375}{86} $

---

Problem 8:

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

Step 1: Expand left side:
$$
0.5 \cdot 3q + 0.5 \cdot 87 = 1.5q + 43.5
$$

So:
$$
1.5q + 43.5 = 1.5q + 43
$$

Subtract $ 1.5q $ from both sides:
$$
43.5 = 43
$$

✘ This is false — contradiction.

So, no solution.

✔ Answer: No solution (inconsistent equation)

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ x = 4 $ |
| 2 | $ x = -10 $ |
| 3 | $ x = 1 $ |
| 4 | $ x = \frac{50}{29} $ |
| 5 | $ x = \frac{9}{10} $ |
| 6 | $ x = 12 $ |
| 7 | $ x = \frac{375}{86} $ |
| 8 | No solution |

Let me know if you'd like these checked or explained further!