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

To solve the linear equations in the provided worksheet, we will go through each section step by step. Here's a detailed explanation and solution for each problem.

---

Section A

#### 1. \(\frac{2x + 5}{3} = 11\)
1. Multiply both sides by 3 to eliminate the denominator:
\[
2x + 5 = 33
\]
2. Subtract 5 from both sides:
\[
2x = 28
\]
3. Divide by 2:
\[
x = 14
\]

#### 2. \(\frac{8 - 3x}{2} = 5\)
1. Multiply both sides by 2:
\[
8 - 3x = 10
\]
2. Subtract 8 from both sides:
\[
-3x = 2
\]
3. Divide by -3:
\[
x = -\frac{2}{3}
\]

#### 3. \(\frac{5 - 9x}{6} = -2\)
1. Multiply both sides by 6:
\[
5 - 9x = -12
\]
2. Subtract 5 from both sides:
\[
-9x = -17
\]
3. Divide by -9:
\[
x = \frac{17}{9}
\]

#### 4. \(\frac{7x + 6}{3} - 9 = -12\)
1. Add 9 to both sides:
\[
\frac{7x + 6}{3} = -3
\]
2. Multiply both sides by 3:
\[
7x + 6 = -9
\]
3. Subtract 6 from both sides:
\[
7x = -15
\]
4. Divide by 7:
\[
x = -\frac{15}{7}
\]

#### 5. \(8x + \frac{1 - 4x}{8} = 7\)
1. Eliminate the fraction by multiplying every term by 8:
\[
64x + (1 - 4x) = 56
\]
2. Simplify:
\[
64x + 1 - 4x = 56
\]
\[
60x + 1 = 56
\]
3. Subtract 1 from both sides:
\[
60x = 55
\]
4. Divide by 60:
\[
x = \frac{55}{60} = \frac{11}{12}
\]

#### 6. \(\frac{5}{x} = -6\)
1. Multiply both sides by \(x\):
\[
5 = -6x
\]
2. Divide by -6:
\[
x = -\frac{5}{6}
\]

#### 7. \(\frac{11}{4x} + 9 = 3\)
1. Subtract 9 from both sides:
\[
\frac{11}{4x} = -6
\]
2. Multiply both sides by \(4x\):
\[
11 = -24x
\]
3. Divide by -24:
\[
x = -\frac{11}{24}
\]

#### 8. \(5 - \frac{3x}{4} = 8x\)
1. Eliminate the fraction by multiplying every term by 4:
\[
20 - 3x = 32x
\]
2. Add \(3x\) to both sides:
\[
20 = 35x
\]
3. Divide by 35:
\[
x = \frac{20}{35} = \frac{4}{7}
\]

#### 9. \(2 + \frac{4x}{3} - 7 = 1\)
1. Simplify the left side:
\[
\frac{4x}{3} - 5 = 1
\]
2. Add 5 to both sides:
\[
\frac{4x}{3} = 6
\]
3. Multiply both sides by 3:
\[
4x = 18
\]
4. Divide by 4:
\[
x = \frac{18}{4} = \frac{9}{2}
\]

#### 10. \(4 - \frac{3x}{2} = 3x + 5\)
1. Eliminate the fraction by multiplying every term by 2:
\[
8 - 3x = 6x + 10
\]
2. Add \(3x\) to both sides:
\[
8 = 9x + 10
\]
3. Subtract 10 from both sides:
\[
-2 = 9x
\]
4. Divide by 9:
\[
x = -\frac{2}{9}
\]

#### 11. \(6 - \frac{2}{x} = 10\)
1. Subtract 6 from both sides:
\[
-\frac{2}{x} = 4
\]
2. Multiply both sides by \(x\):
\[
-2 = 4x
\]
3. Divide by 4:
\[
x = -\frac{1}{2}
\]

#### 12. \(4 - \frac{2x}{9} + x = -1\)
1. Combine like terms:
\[
4 + \left(x - \frac{2x}{9}\right) = -1
\]
\[
4 + \frac{9x - 2x}{9} = -1
\]
\[
4 + \frac{7x}{9} = -1
\]
2. Subtract 4 from both sides:
\[
\frac{7x}{9} = -5
\]
3. Multiply both sides by 9:
\[
7x = -45
\]
4. Divide by 7:
\[
x = -\frac{45}{7}
\]

---

Section B

#### 1. \(4(2x - 3) = 8(2x + 5)\)
1. Distribute on both sides:
\[
8x - 12 = 16x + 40
\]
2. Subtract \(8x\) from both sides:
\[
-12 = 8x + 40
\]
3. Subtract 40 from both sides:
\[
-52 = 8x
\]
4. Divide by 8:
\[
x = -\frac{52}{8} = -\frac{13}{2}
\]

#### 2. \(3(4x - 5) = 5(2x - 5)\)
1. Distribute on both sides:
\[
12x - 15 = 10x - 25
\]
2. Subtract \(10x\) from both sides:
\[
2x - 15 = -25
\]
3. Add 15 to both sides:
\[
2x = -10
\]
4. Divide by 2:
\[
x = -5
\]

#### 3. \(8(6x + 2) = 5(x - 2)\)
1. Distribute on both sides:
\[
48x + 16 = 5x - 10
\]
2. Subtract \(5x\) from both sides:
\[
43x + 16 = -10
\]
3. Subtract 16 from both sides:
\[
43x = -26
\]
4. Divide by 43:
\[
x = -\frac{26}{43}
\]

#### 4. \(2(3x - 4) = 7(11 - 2x)\)
1. Distribute on both sides:
\[
6x - 8 = 77 - 14x
\]
2. Add \(14x\) to both sides:
\[
20x - 8 = 77
\]
3. Add 8 to both sides:
\[
20x = 85
\]
4. Divide by 20:
\[
x = \frac{85}{20} = \frac{17}{4}
\]

#### 5. \(7(5 - x) = -4(x - 11)\)
1. Distribute on both sides:
\[
35 - 7x = -4x + 44
\]
2. Add \(4x\) to both sides:
\[
35 - 3x = 44
\]
3. Subtract 35 from both sides:
\[
-3x = 9
\]
4. Divide by -3:
\[
x = -3
\]

#### 6. \(-4(x - 8) = -6(4 + 3x)\)
1. Distribute on both sides:
\[
-4x + 32 = -24 - 18x
\]
2. Add \(18x\) to both sides:
\[
14x + 32 = -24
\]
3. Subtract 32 from both sides:
\[
14x = -56
\]
4. Divide by 14:
\[
x = -4
\]

#### 7. \(7(4 - 3x) = 2(8x - 9) + 6\)
1. Distribute on both sides:
\[
28 - 21x = 16x - 18 + 6
\]
\[
28 - 21x = 16x - 12
\]
2. Add \(21x\) to both sides:
\[
28 = 37x - 12
\]
3. Add 12 to both sides:
\[
40 = 37x
\]
4. Divide by 37:
\[
x = \frac{40}{37}
\]

#### 8. \(-6(3 - 4x) + 2x = 8(x + 11)\)
1. Distribute on both sides:
\[
-18 + 24x + 2x = 8x + 88
\]
\[
-18 + 26x = 8x + 88
\]
2. Subtract \(8x\) from both sides:
\[
-18 + 18x = 88
\]
3. Add 18 to both sides:
\[
18x = 106
\]
4. Divide by 18:
\[
x = \frac{106}{18} = \frac{53}{9}
\]

#### 9. \(3(2x - 6) = 3 - 4(3 - x)\)
1. Distribute on both sides:
\[
6x - 18 = 3 - 12 + 4x
\]
\[
6x - 18 = -9 + 4x
\]
2. Subtract \(4x\) from both sides:
\[
2x - 18 = -9
\]
3. Add 18 to both sides:
\[
2x = 9
\]
4. Divide by 2:
\[
x = \frac{9}{2}
\]

#### 10. \(9(2x - 1) - 3x = 3(12 + x)\)
1. Distribute on both sides:
\[
18x - 9 - 3x = 36 + 3x
\]
\[
15x - 9 = 36 + 3x
\]
2. Subtract \(3x\) from both sides:
\[
12x - 9 = 36
\]
3. Add 9 to both sides:
\[
12x = 45
\]
4. Divide by 12:
\[
x = \frac{45}{12} = \frac{15}{4}
\]

#### 11. \(4x - (2x - 8) = 5(1 + 2x)\)
1. Simplify the left side:
\[
4x - 2x + 8 = 5(1 + 2x)
\]
\[
2x + 8 = 5 + 10x
\]
2. Subtract \(2x\) from both sides:
\[
8 = 5 + 8x
\]
3. Subtract 5 from both sides:
\[
3 = 8x
\]
4. Divide by 8:
\[
x = \frac{3}{8}
\]

#### 12. \(10 - 6(8x - 2) = 9x - (3 + 4x)\)
1. Distribute on both sides:
\[
10 - 48x + 12 = 9x - 3 - 4x
\]
\[
22 - 48x = 5x - 3
\]
2. Add \(48x\) to both sides:
\[
22 = 53x - 3
\]
3. Add 3 to both sides:
\[
25 = 53x
\]
4. Divide by 53:
\[
x = \frac{25}{53}
\]

---

Section C

#### 1. \(\frac{5x - 2}{3} = \frac{4x + 1}{2}\)
1. Eliminate the fractions by cross-multiplying:
\[
2(5x - 2) = 3(4x + 1)
\]
2. Distribute on both sides:
\[
10x - 4 = 12x + 3
\]
3. Subtract \(10x\) from both sides:
\[
-4 = 2x + 3
\]
4. Subtract 3 from both sides:
\[
-7 = 2x
\]
5. Divide by 2:
\[
x = -\frac{7}{2}
\]

#### 2. \(\frac{7x - 8}{5} = \frac{2x + 5}{4}\)
1. Eliminate the fractions by cross-multiplying:
\[
4(7x - 8) = 5(2x + 5)
\]
2. Distribute on both sides:
\[
28x - 32 = 10x + 25
\]
3. Subtract \(10x\) from both sides:
\[
18x - 32 = 25
\]
4. Add 32 to both sides:
\[
18x = 57
\]
5. Divide by 18:
\[
x = \frac{57}{18} = \frac{19}{6}
\]

#### 3. \(\frac{-8x - 1}{2} = \frac{5 - 3x}{6}\)
1. Eliminate the fractions by cross-multiplying:
\[
6(-8x - 1) = 2(5 - 3x)
\]
2. Distribute on both sides:
\[
-48x - 6 = 10 - 6x
\]
3. Add \(48x\) to both sides:
\[
-6 = 42x + 10
\]
4. Subtract 10 from both sides:
\[
-16 = 42x
\]
5. Divide by 42:
\[
x = -\frac{16}{42} = -\frac{8}{21}
\]

#### 4. \(\frac{5(x + 11)}{3} = \frac{3(1 + x)}{2}\)
1. Eliminate the fractions by cross-multiplying:
\[
2 \cdot 5(x + 11) = 3 \cdot 3(1 + x)
\]
\[
10(x + 11) = 9(1 + x)
\]
2. Distribute on both sides:
\[
10x + 110 = 9 + 9x
\]
3. Subtract \(9x\) from both sides:
\[
x + 110 = 9
\]
4. Subtract 110 from both sides:
\[
x = -101
\]

#### 5. \(\frac{3(2 + 5x)}{4} = \frac{2(6x - 3)}{5}\)
1. Eliminate the fractions by cross-multiplying:
\[
5 \cdot 3(2 + 5x) = 4 \cdot 2(6x - 3)
\]
\[
15(2 + 5x) = 8(6x - 3)
\]
2. Distribute on both sides:
\[
30 + 75x = 48x - 24
\]
3. Subtract \(48x\) from both sides:
\[
30 + 27x = -24
\]
4. Subtract 30 from both sides:
\[
27x = -54
\]
5. Divide by 27:
\[
x = -2
\]

#### 6. \(\frac{2(3x - 5)}{3} = \frac{-4(x - 2)}{7}\)
1. Eliminate the fractions by cross-multiplying:
\[
7 \cdot 2(3x - 5) = 3 \cdot (-4)(x - 2)
\]
\[
14(3x - 5) = -12(x - 2)
\]
2. Distribute on both sides:
\[
42x - 70 = -12x + 24
\]
3. Add \(12x\) to both sides:
\[
54x - 70 = 24
\]
4. Add 70 to both sides:
\[
54x = 94
\]
5. Divide by 54:
\[
x = \frac{94}{54} = \frac{47}{27}
\]

#### 7. \(\frac{1}{2}(2x - 6) = \frac{1}{4}(8 - 12x)\)
1. Simplify both sides:
\[
x - 3 = 2 - 3x
\]
2. Add \(3x\) to both sides:
\[
4x - 3 = 2
\]
3. Add 3 to both sides:
\[
4x = 5
\]
4. Divide by 4:
\[
x = \frac{5}{4}
\]

#### 8. \(\frac{1}{2}(5x + 7) = \frac{3}{4}(3x - 1)\)
1. Eliminate the fractions by multiplying every term by 4:
\[
2(5x + 7) = 3(3x - 1)
\]
2. Distribute on both sides:
\[
10x + 14 = 9x - 3
\]
3. Subtract \(9x\) from both sides:
\[
x + 14 = -3
\]
4. Subtract 14 from both sides:
\[
x = -17
\]

#### 9. \(\frac{5}{3x + 1} = 12\)
1. Multiply both sides by \(3x + 1\):
\[
5 = 12(3x + 1)
\]
2. Distribute on the right side:
\[
5 = 36x + 12
\]
3. Subtract 12 from both sides:
\[
-7 = 36x
\]
4. Divide by 36:
\[
x = -\frac{7}{36}
\]

#### 10. \(\frac{x + 2}{x + 3} = 4\)
1. Multiply both sides by \(x + 3\):
\[
x + 2 = 4(x + 3)
\]
2. Distribute on the right side:
\[
x + 2 = 4x + 12
\]
3. Subtract \(x\) from both sides:
\[
2 = 3x + 12
\]
4. Subtract 12 from both sides:
\[
-10 = 3x
\]
5. Divide by 3:
\[
x = -\frac{10}{3}
\]

#### 11. \(\frac{2x - 9}{3x - 2} = -3\)
1. Multiply both sides by \(3x - 2\):
\[
2x - 9 = -3(3x - 2)
\]
2. Distribute on the right side:
\[
2x - 9 = -9x + 6
\]
3. Add \(9x\) to both sides:
\[
11x - 9 = 6
\]
4. Add 9 to both sides:
\[
11x = 15
\]
5. Divide by 11:
\[
x = \frac{15}{11}
\]

#### 12. \(\frac{2}{3x + 10} = \frac{1}{x - 1}\)
1. Cross-multiply:
\[
2(x - 1) = 1(3x + 10)
\]
2. Distribute on both sides:
\[
2x - 2 = 3x + 10
\]
3. Subtract \(2x\) from both sides:
\[
-2 = x + 10
\]
4. Subtract 10 from both sides:
\[
x = -12
\]

#### 13. \(\frac{2}{7x + 3} = \frac{9}{2x - 5}\)
1. Cross-multiply:
\[
2(2x - 5) = 9(7x + 3)
\]
2. Distribute on both sides:
\[
4x - 10 = 63x + 27
\]
3. Subtract \(4x\) from both sides:
\[
-10 = 59x + 27
\]
4. Subtract 27 from both sides:
\[
-37 = 59x
\]
5. Divide by 59:
\[
x = -\frac{37}{59}
\]

#### 14. \(\frac{8}{6x + 12} = -\frac{11}{7x - 10}\)
1. Cross-multiply:
\[
8(7x - 10) = -11(6x + 12)
\]
2. Distribute on both sides:
\[
56x - 80 = -66x - 132
\]
3. Add \(66x\) to both sides:
\[
122x - 80 = -132
\]
4. Add 80 to both sides:
\[
122x = -52
\]
5. Divide by 122:
\[
x = -\frac{52}{122} = -\frac{26}{61}
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{Section A:} & 1. 14, 2. -\frac{2}{3}, 3. \frac{17}{9}, 4. -\frac{15}{7}, 5. \frac{11}{12}, 6. -\frac{5}{6}, 7. -\frac{11}{24}, 8. \frac{4}{7}, 9. \frac{9}{2}, 10. -\frac{2}{9}, 11. -\frac{1}{2}, 12. -\frac{45}{7} \\
\text{Section B:} & 1. -\frac{13}{2}, 2. -5, 3. -\frac{26}{43}, 4. \frac{17}{4}, 5. -3, 6. -4, 7. \frac{40}{37}, 8. \frac{53}{9}, 9. \frac{9}{2}, 10. \frac{15}{4}, 11. \frac{3}{8}, 12. \frac{25}{53} \\
\text{Section C:} & 1. -\frac{7}{2}, 2. \frac{19}{6}, 3. -\frac{8}{21}, 4. -101, 5. -2, 6. \frac{47}{27}, 7. \frac{5}{4}, 8. -17, 9. -\frac{7}{36}, 10. -\frac{10}{3}, 11. \frac{15}{11}, 12. -12, 13. -\frac{37}{59}, 14. -\frac{26}{61}
\end{array}
}
\]