Free worksheets for linear equations (grades 6-9, pre-algebra - Free Printable

Problem Overview:

The task is to solve a series of linear equations to find the value of \( x \). The worksheet is divided into six sections (A, B, C, D, E, F), each containing different types of linear equations. We will solve each equation step by step.

---

Section A: Simple Addition Equations

These equations are of the form \( x + a = b \). To solve for \( x \), subtract \( a \) from both sides.

#### Solutions:
1. \( x + 3 = 11 \)
\[
x = 11 - 3 = 8
\]
2. \( x + 2 = 8 \)
\[
x = 8 - 2 = 6
\]
3. \( x + 5 = 7 \)
\[
x = 7 - 5 = 2
\]
4. \( x + 7 = 13 \)
\[
x = 13 - 7 = 6
\]
5. \( x + 4 = 14 \)
\[
x = 14 - 4 = 10
\]
6. \( x + 7 = 9 \)
\[
x = 9 - 7 = 2
\]
7. \( x + 3 = 9 \)
\[
x = 9 - 3 = 6
\]
8. \( x + 12 = 17 \)
\[
x = 17 - 12 = 5
\]
9. \( x + 6 = 24 \)
\[
x = 24 - 6 = 18
\]
10. \( x + 5 = 36 \)
\[
x = 36 - 5 = 31
\]
11. \( x + 8 = 43 \)
\[
x = 43 - 8 = 35
\]
12. \( x + 9 = 61 \)
\[
x = 61 - 9 = 52
\]

---

Section B: Simple Addition Equations (Reversed Order)

These equations are of the form \( a + x = b \). To solve for \( x \), subtract \( a \) from both sides.

#### Solutions:
1. \( 4 + x = 6 \)
\[
x = 6 - 4 = 2
\]
2. \( 2 + x = 7 \)
\[
x = 7 - 2 = 5
\]
3. \( 8 + x = 11 \)
\[
x = 11 - 8 = 3
\]
4. \( 5 + x = 9 \)
\[
x = 9 - 5 = 4
\]
5. \( 7 + x = 12 \)
\[
x = 12 - 7 = 5
\]
6. \( 12 + x = 18 \)
\[
x = 18 - 12 = 6
\]
7. \( 14 + x = 23 \)
\[
x = 23 - 14 = 9
\]
8. \( 19 + x = 32 \)
\[
x = 32 - 19 = 13
\]
9. \( 7 + x = 40 \)
\[
x = 40 - 7 = 33
\]
10. \( 8 + x = 72 \)
\[
x = 72 - 8 = 64
\]
11. \( 11 + x = 64 \)
\[
x = 64 - 11 = 53
\]
12. \( 28 + x = 90 \)
\[
x = 90 - 28 = 62
\]

---

Section C: Simple Subtraction Equations

These equations are of the form \( x - a = b \). To solve for \( x \), add \( a \) to both sides.

#### Solutions:
1. \( x - 4 = 7 \)
\[
x = 7 + 4 = 11
\]
2. \( x - 6 = 4 \)
\[
x = 4 + 6 = 10
\]
3. \( x - 1 = 6 \)
\[
x = 6 + 1 = 7
\]
4. \( x - 7 = 13 \)
\[
x = 13 + 7 = 20
\]
5. \( x - 10 = 2 \)
\[
x = 2 + 10 = 12
\]
6. \( x - 7 = 18 \)
\[
x = 18 + 7 = 25
\]
7. \( x - 11 = 8 \)
\[
x = 8 + 11 = 19
\]
8. \( x - 5 = 16 \)
\[
x = 16 + 5 = 21
\]
9. \( x - 9 = 25 \)
\[
x = 25 + 9 = 34
\]
10. \( x - 12 = 31 \)
\[
x = 31 + 12 = 43
\]
11. \( x - 16 = 29 \)
\[
x = 29 + 16 = 45
\]
12. \( x - 28 = 78 \)
\[
x = 78 + 28 = 106
\]

---

Section D: Multiplication Equations

These equations are of the form \( ax = b \). To solve for \( x \), divide both sides by \( a \).

#### Solutions:
1. \( 2x = 6 \)
\[
x = \frac{6}{2} = 3
\]
2. \( 5x = 10 \)
\[
x = \frac{10}{5} = 2
\]
3. \( 4x = 12 \)
\[
x = \frac{12}{4} = 3
\]
4. \( 10x = 90 \)
\[
x = \frac{90}{10} = 9
\]
5. \( 3x = 15 \)
\[
x = \frac{15}{3} = 5
\]
6. \( 6x = 24 \)
\[
x = \frac{24}{6} = 4
\]
7. \( 7x = 35 \)
\[
x = \frac{35}{7} = 5
\]
8. \( 12x = 36 \)
\[
x = \frac{36}{12} = 3
\]
9. \( 15x = 30 \)
\[
x = \frac{30}{15} = 2
\]
10. \( 20x = 40 \)
\[
x = \frac{40}{20} = 2
\]
11. \( 40x = 120 \)
\[
x = \frac{120}{40} = 3
\]
12. \( 50x = 200 \)
\[
x = \frac{200}{50} = 4
\]

---

Section E: Division Equations

These equations are of the form \( \frac{x}{a} = b \). To solve for \( x \), multiply both sides by \( a \).

#### Solutions:
1. \( \frac{x}{3} = 4 \)
\[
x = 4 \times 3 = 12
\]
2. \( \frac{x}{2} = 8 \)
\[
x = 8 \times 2 = 16
\]
3. \( \frac{x}{5} = 7 \)
\[
x = 7 \times 5 = 35
\]
4. \( \frac{x}{8} = 4 \)
\[
x = 4 \times 8 = 32
\]
5. \( \frac{x}{7} = 3 \)
\[
x = 3 \times 7 = 21
\]
6. \( \frac{x}{5} = 4 \)
\[
x = 4 \times 5 = 20
\]
7. \( \frac{x}{2} = 9 \)
\[
x = 9 \times 2 = 18
\]
8. \( \frac{x}{9} = 5 \)
\[
x = 5 \times 9 = 45
\]
9. \( \frac{x}{7} = 8 \)
\[
x = 8 \times 7 = 56
\]
10. \( \frac{x}{12} = 6 \)
\[
x = 6 \times 12 = 72
\]
11. \( \frac{x}{14} = 2 \)
\[
x = 2 \times 14 = 28
\]
12. \( \frac{x}{30} = 5 \)
\[
x = 5 \times 30 = 150
\]

---

Section F: Mixed Equations

These equations involve addition, subtraction, multiplication, and division. Solve step by step.

#### Solutions:
1. \( 4x = 48 \)
\[
x = \frac{48}{4} = 12
\]
2. \( x + 13 = 22 \)
\[
x = 22 - 13 = 9
\]
3. \( 9x = 63 \)
\[
x = \frac{63}{9} = 7
\]
4. \( 11x = 132 \)
\[
x = \frac{132}{11} = 12
\]
5. \( 12 + x = 26 \)
\[
x = 26 - 12 = 14
\]
6. \( \frac{x}{8} = 12 \)
\[
x = 12 \times 8 = 96
\]
7. \( x - 19 = 30 \)
\[
x = 30 + 19 = 49
\]
8. \( 10x = 160 \)
\[
x = \frac{160}{10} = 16
\]
9. \( 13 + x = 27 \)
\[
x = 27 - 13 = 14
\]
10. \( 6x = 42 \)
\[
x = \frac{42}{6} = 7
\]
11. \( x + 17 = 42 \)
\[
x = 42 - 17 = 25
\]
12. \( \frac{x}{11} = 11 \)
\[
x = 11 \times 11 = 121
\]
13. \( 7x = 56 \)
\[
x = \frac{56}{7} = 8
\]
14. \( 18 + x = 24 \)
\[
x = 24 - 18 = 6
\]
15. \( \frac{x}{4} = 12 \)
\[
x = 12 \times 4 = 48
\]
16. \( 25 + x = 39 \)
\[
x = 39 - 25 = 14
\]
17. \( 5x = 100 \)
\[
x = \frac{100}{5} = 20
\]
18. \( \frac{x}{3} = 300 \)
\[
x = 300 \times 3 = 900
\]
19. \( x + 49 = 110 \)
\[
x = 110 - 49 = 61
\]
20. \( 100x = 6500 \)
\[
x = \frac{6500}{100} = 65
\]

---

Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Section A: } 8, 6, 2, 6, 10, 2, 6, 5, 18, 31, 35, 52 \\
\text{Section B: } 2, 5, 3, 4, 5, 6, 9, 13, 33, 64, 53, 62 \\
\text{Section C: } 11, 10, 7, 20, 12, 25, 19, 21, 34, 43, 45, 106 \\
\text{Section D: } 3, 2, 3, 9, 5, 4, 5, 3, 2, 2, 3, 4 \\
\text{Section E: } 12, 16, 35, 32, 21, 20, 18, 45, 56, 72, 28, 150 \\
\text{Section F: } 12, 9, 7, 12, 14, 96, 49, 16, 14, 7, 25, 121, 8, 6, 48, 14, 20, 900, 61, 65
\end{array}
}
\]