Algebra. Level 7. Equations. Solving Linear Equations (D) | PDF ... - Free Printable

Let's solve the equations from Section A, Section B, and Section C step by step, showing all work and simplifying answers as required.

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🔷 Section A: Solving Linear Equations

We'll solve each equation one at a time.

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#### 1) $\frac{2x + 5}{3} = 11$

Multiply both sides by 3:
$$
2x + 5 = 33
$$
Subtract 5:
$$
2x = 28 \Rightarrow x = 14
$$

✔ $x = 14$

---

#### 2) $\frac{8 - 3x}{2} = 5$

Multiply both sides by 2:
$$
8 - 3x = 10
$$
Subtract 8:
$$
-3x = 2 \Rightarrow x = -\frac{2}{3}
$$

✔ $x = -\frac{2}{3}$

---

#### 3) $\frac{5 - 9x}{6} = -2$

Multiply both sides by 6:
$$
5 - 9x = -12
$$
Subtract 5:
$$
-9x = -17 \Rightarrow x = \frac{17}{9}
$$

✔ $x = \frac{17}{9}$

---

#### 4) $\frac{7x + 6}{3} - 9 = -12$

Add 9 to both sides:
$$
\frac{7x + 6}{3} = -3
$$
Multiply by 3:
$$
7x + 6 = -9
$$
Subtract 6:
$$
7x = -15 \Rightarrow x = -\frac{15}{7}
$$

✔ $x = -\frac{15}{7}$

---

#### 5) $8x + \frac{1 - 4x}{8} = 7$

Multiply entire equation by 8 to eliminate denominator:
$$
64x + (1 - 4x) = 56
$$
Simplify:
$$
64x + 1 - 4x = 56 \Rightarrow 60x + 1 = 56
$$
Subtract 1:
$$
60x = 55 \Rightarrow x = \frac{55}{60} = \frac{11}{12}
$$

✔ $x = \frac{11}{12}$

---

#### 6) $\frac{5}{x} = -6$

Multiply both sides by $x$:
$$
5 = -6x \Rightarrow x = -\frac{5}{6}
$$

✔ $x = -\frac{5}{6}$

---

#### 7) $\frac{11}{4x} + 9 = 3$

Subtract 9:
$$
\frac{11}{4x} = -6
$$
Multiply both sides by $4x$:
$$
11 = -24x \Rightarrow x = -\frac{11}{24}
$$

✔ $x = -\frac{11}{24}$

---

#### 8) $5 - \frac{3x}{4} = 8x$

Multiply all terms by 4:
$$
20 - 3x = 32x
$$
Add $3x$:
$$
20 = 35x \Rightarrow x = \frac{20}{35} = \frac{4}{7}
$$

✔ $x = \frac{4}{7}$

---

#### 9) $2 + \frac{4x}{3} - 7 = 1$

Simplify left side:
$$
\frac{4x}{3} - 5 = 1
$$
Add 5:
$$
\frac{4x}{3} = 6
$$
Multiply by 3:
$$
4x = 18 \Rightarrow x = \frac{9}{2}
$$

✔ $x = \frac{9}{2}$

---

#### 10) $4 - \frac{3x}{2} = 3x + 5$

Multiply all terms by 2:
$$
8 - 3x = 6x + 10
$$
Move variables:
$$
8 - 10 = 6x + 3x \Rightarrow -2 = 9x \Rightarrow x = -\frac{2}{9}
$$

✔ $x = -\frac{2}{9}$

---

#### 11) $6 - \frac{2}{x} = 10$

Subtract 6:
$$
-\frac{2}{x} = 4
$$
Multiply both sides by $x$:
$$
-2 = 4x \Rightarrow x = -\frac{1}{2}
$$

✔ $x = -\frac{1}{2}$

---

#### 12) $4 - \frac{2x}{9} + x = -1$

Combine like terms:
$$
4 + x - \frac{2x}{9} = -1
\Rightarrow 4 + \left(1 - \frac{2}{9}\right)x = -1
\Rightarrow 4 + \frac{7x}{9} = -1
$$
Subtract 4:
$$
\frac{7x}{9} = -5
$$
Multiply by 9:
$$
7x = -45 \Rightarrow x = -\frac{45}{7}
$$

✔ $x = -\frac{45}{7}$

---

✔ Section A Answers:

| Q | Answer |
|---|----------------|
| 1 | 14 |
| 2 | $-\frac{2}{3}$ |
| 3 | $\frac{17}{9}$ |
| 4 | $-\frac{15}{7}$ |
| 5 | $\frac{11}{12}$ |
| 6 | $-\frac{5}{6}$ |
| 7 | $-\frac{11}{24}$ |
| 8 | $\frac{4}{7}$ |
| 9 | $\frac{9}{2}$ |
|10 | $-\frac{2}{9}$ |
|11 | $-\frac{1}{2}$ |
|12 | $-\frac{45}{7}$ |

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🔷 Section B: Equations with Distribution

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#### 1) $4(2x - 3) = 8(2x + 5)$

Expand:
$$
8x - 12 = 16x + 40
$$
Move terms:
$$
-12 - 40 = 16x - 8x \Rightarrow -52 = 8x \Rightarrow x = -\frac{13}{2}
$$

✔ $x = -\frac{13}{2}$

---

#### 2) $3(4x - 5) = 5(2x - 5)$

$$
12x - 15 = 10x - 25
\Rightarrow 12x - 10x = -25 + 15 \Rightarrow 2x = -10 \Rightarrow x = -5
$$

✔ $x = -5$

---

#### 3) $8(6x + 2) = 5(x - 2)$

$$
48x + 16 = 5x - 10
\Rightarrow 48x - 5x = -10 - 16 \Rightarrow 43x = -26 \Rightarrow x = -\frac{26}{43}
$$

✔ $x = -\frac{26}{43}$

---

#### 4) $2(3x - 4) = 7(11 - 2x)$

$$
6x - 8 = 77 - 14x
\Rightarrow 6x + 14x = 77 + 8 \Rightarrow 20x = 85 \Rightarrow x = \frac{17}{4}
$$

✔ $x = \frac{17}{4}$

---

#### 5) $7(5 - x) = -4(x - 11)$

$$
35 - 7x = -4x + 44
\Rightarrow -7x + 4x = 44 - 35 \Rightarrow -3x = 9 \Rightarrow x = -3
$$

✔ $x = -3$

---

#### 6) $-4(x - 8) = -6(4 + 3x)$

$$
-4x + 32 = -24 - 18x
\Rightarrow -4x + 18x = -24 - 32 \Rightarrow 14x = -56 \Rightarrow x = -4
$$

✔ $x = -4$

---

#### 7) $7(4 - 3x) = 2(8x - 9) + 6$

Left: $28 - 21x$

Right: $16x - 18 + 6 = 16x - 12$

So:
$$
28 - 21x = 16x - 12
\Rightarrow 28 + 12 = 16x + 21x \Rightarrow 40 = 37x \Rightarrow x = \frac{40}{37}
$$

✔ $x = \frac{40}{37}$

---

#### 8) $-6(3 - 4x) + 2x = 8(x + 11)$

Left: $-18 + 24x + 2x = 26x - 18$

Right: $8x + 88$

So:
$$
26x - 18 = 8x + 88
\Rightarrow 26x - 8x = 88 + 18 \Rightarrow 18x = 106 \Rightarrow x = \frac{53}{9}
$$

✔ $x = \frac{53}{9}$

---

#### 9) $3(2x - 6) = 3 - 4(3 - x)$

Left: $6x - 18$

Right: $3 - 12 + 4x = 4x - 9$

So:
$$
6x - 18 = 4x - 9
\Rightarrow 6x - 4x = -9 + 18 \Rightarrow 2x = 9 \Rightarrow x = \frac{9}{2}
$$

✔ $x = \frac{9}{2}$

---

#### 10) $9(2x - 1) - 3x = 3(12 + x)$

Left: $18x - 9 - 3x = 15x - 9$

Right: $36 + 3x$

So:
$$
15x - 9 = 36 + 3x
\Rightarrow 15x - 3x = 36 + 9 \Rightarrow 12x = 45 \Rightarrow x = \frac{15}{4}
$$

✔ $x = \frac{15}{4}$

---

#### 11) $4x - (2x - 8) = 5(1 + 2x)$

Left: $4x - 2x + 8 = 2x + 8$

Right: $5 + 10x$

So:
$$
2x + 8 = 5 + 10x
\Rightarrow 8 - 5 = 10x - 2x \Rightarrow 3 = 8x \Rightarrow x = \frac{3}{8}
$$

✔ $x = \frac{3}{8}$

---

#### 12) $10 - 6(8x - 2) = 9x - (3 + 4x)$

Left: $10 - 48x + 12 = 22 - 48x$

Right: $9x - 3 - 4x = 5x - 3$

So:
$$
22 - 48x = 5x - 3
\Rightarrow 22 + 3 = 5x + 48x \Rightarrow 25 = 53x \Rightarrow x = \frac{25}{53}
$$

✔ $x = \frac{25}{53}$

---

✔ Section B Answers:

| Q | Answer |
|---|----------------|
| 1 | $-\frac{13}{2}$ |
| 2 | $-5$ |
| 3 | $-\frac{26}{43}$ |
| 4 | $\frac{17}{4}$ |
| 5 | $-3$ |
| 6 | $-4$ |
| 7 | $\frac{40}{37}$ |
| 8 | $\frac{53}{9}$ |
| 9 | $\frac{9}{2}$ |
|10 | $\frac{15}{4}$ |
|11 | $\frac{3}{8}$ |
|12 | $\frac{25}{53}$ |

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🔷 Section C: More Complex Fractions & Equations

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#### 1) $\frac{5x - 2}{3} = \frac{4x + 1}{2}$

Cross-multiply:
$$
2(5x - 2) = 3(4x + 1)
\Rightarrow 10x - 4 = 12x + 3
\Rightarrow -4 - 3 = 12x - 10x \Rightarrow -7 = 2x \Rightarrow x = -\frac{7}{2}
$$

✔ $x = -\frac{7}{2}$

---

#### 2) $\frac{7x - 8}{5} = \frac{2x + 5}{4}$

Cross-multiply:
$$
4(7x - 8) = 5(2x + 5)
\Rightarrow 28x - 32 = 10x + 25
\Rightarrow 28x - 10x = 25 + 32 \Rightarrow 18x = 57 \Rightarrow x = \frac{19}{6}
$$

✔ $x = \frac{19}{6}$

---

#### 3) $\frac{-8x - 1}{2} = \frac{5 - 3x}{6}$

Multiply both sides by 6:
$$
3(-8x - 1) = 5 - 3x
\Rightarrow -24x - 3 = 5 - 3x
\Rightarrow -24x + 3x = 5 + 3 \Rightarrow -21x = 8 \Rightarrow x = -\frac{8}{21}
$$

✔ $x = -\frac{8}{21}$

---

#### 4) $\frac{5(x + 11)}{3} = \frac{3(1 + x)}{2}$

Multiply both sides by 6:
$$
2 \cdot 5(x + 11) = 3 \cdot 3(1 + x)
\Rightarrow 10(x + 11) = 9(1 + x)
\Rightarrow 10x + 110 = 9 + 9x
\Rightarrow 10x - 9x = 9 - 110 \Rightarrow x = -101
$$

✔ $x = -101$

---

#### 5) $\frac{3(2 + 5x)}{4} = \frac{2(6x - 3)}{5}$

Cross-multiply:
$$
5 \cdot 3(2 + 5x) = 4 \cdot 2(6x - 3)
\Rightarrow 15(2 + 5x) = 8(6x - 3)
\Rightarrow 30 + 75x = 48x - 24
\Rightarrow 75x - 48x = -24 - 30 \Rightarrow 27x = -54 \Rightarrow x = -2
$$

✔ $x = -2$

---

#### 6) $\frac{2(3x - 5)}{3} = \frac{-4(x - 2)}{7}$

Cross-multiply:
$$
7 \cdot 2(3x - 5) = 3 \cdot (-4)(x - 2)
\Rightarrow 14(3x - 5) = -12(x - 2)
\Rightarrow 42x - 70 = -12x + 24
\Rightarrow 42x + 12x = 24 + 70 \Rightarrow 54x = 94 \Rightarrow x = \frac{47}{27}
$$

✔ $x = \frac{47}{27}$

---

#### 7) $\frac{1}{2}(2x - 6) = \frac{1}{4}(8 - 12x)$

Simplify both sides:
Left: $x - 3$

Right: $2 - 3x$

So:
$$
x - 3 = 2 - 3x
\Rightarrow x + 3x = 2 + 3 \Rightarrow 4x = 5 \Rightarrow x = \frac{5}{4}
$$

✔ $x = \frac{5}{4}$

---

#### 8) $\frac{1}{2}(5x + 7) = \frac{3}{4}(3x - 1)$

Multiply both sides by 4:
$$
2(5x + 7) = 3(3x - 1)
\Rightarrow 10x + 14 = 9x - 3
\Rightarrow 10x - 9x = -3 - 14 \Rightarrow x = -17
$$

✔ $x = -17$

---

#### 9) $\frac{5}{3x + 1} = 12$

Multiply both sides by $3x + 1$:
$$
5 = 12(3x + 1)
\Rightarrow 5 = 36x + 12
\Rightarrow 5 - 12 = 36x \Rightarrow -7 = 36x \Rightarrow x = -\frac{7}{36}
$$

✔ $x = -\frac{7}{36}$

---

#### 10) $\frac{x + 2}{x + 3} = 4$

Multiply both sides by $x + 3$:
$$
x + 2 = 4(x + 3)
\Rightarrow x + 2 = 4x + 12
\Rightarrow 2 - 12 = 4x - x \Rightarrow -10 = 3x \Rightarrow x = -\frac{10}{3}
$$

✔ $x = -\frac{10}{3}$

---

#### 11) $\frac{2x - 9}{3x - 2} = -3$

Multiply both sides by $3x - 2$:
$$
2x - 9 = -3(3x - 2)
\Rightarrow 2x - 9 = -9x + 6
\Rightarrow 2x + 9x = 6 + 9 \Rightarrow 11x = 15 \Rightarrow x = \frac{15}{11}
$$

✔ $x = \frac{15}{11}$

---

#### 12) $\frac{2}{3x + 10} = \frac{1}{x - 1}$

Cross-multiply:
$$
2(x - 1) = 1(3x + 10)
\Rightarrow 2x - 2 = 3x + 10
\Rightarrow -2 - 10 = 3x - 2x \Rightarrow -12 = x
$$

✔ $x = -12$

---

#### 13) $\frac{2}{7x + 3} = \frac{9}{2x - 5}$

Cross-multiply:
$$
2(2x - 5) = 9(7x + 3)
\Rightarrow 4x - 10 = 63x + 27
\Rightarrow 4x - 63x = 27 + 10 \Rightarrow -59x = 37 \Rightarrow x = -\frac{37}{59}
$$

✔ $x = -\frac{37}{59}$

---

#### 14) $\frac{8}{6x + 12} = -\frac{11}{7x - 10}$

Note: $6x + 12 = 6(x + 2)$

Cross-multiply:
$$
8(7x - 10) = -11(6x + 12)
\Rightarrow 56x - 80 = -66x - 132
\Rightarrow 56x + 66x = -132 + 80 \Rightarrow 122x = -52 \Rightarrow x = -\frac{26}{61}
$$

✔ $x = -\frac{26}{61}$

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✔ Section C Answers:

| Q | Answer |
|---|-------------------|
| 1 | $-\frac{7}{2}$ |
| 2 | $\frac{19}{6}$ |
| 3 | $-\frac{8}{21}$ |
| 4 | $-101$ |
| 5 | $-2$ |
| 6 | $\frac{47}{27}$ |
| 7 | $\frac{5}{4}$ |
| 8 | $-17$ |
| 9 | $-\frac{7}{36}$ |
|10 | $-\frac{10}{3}$ |
|11 | $\frac{15}{11}$ |
|12 | $-12$ |
|13 | $-\frac{37}{59}$ |
|14 | $-\frac{26}{61}$ |

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📌 Final Notes:

- All answers are simplified fractions or decimals as requested.
- For fractions, we reduced them to lowest terms.
- For equations involving denominators, we checked for restrictions (e.g., $x \neq -3$ in Q10), but no invalid solutions arose.
- The worksheet is designed for Grade C (Level 7) — this matches the difficulty level of the problems.

Let me know if you'd like these answers formatted into a printable answer key!