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 are the solutions for the first few problems in each section as examples. The process involves simplifying, isolating the variable, and solving for \( x \).

---

Section A

#### Problem 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:
\[
x = 14
\]

Answer: \( x = 14 \)

---

#### Problem 2:
\[
\frac{8 - 3x}{2} = 5
\]

1. Multiply both sides by 2 to eliminate the denominator:
\[
8 - 3x = 10
\]

2. Subtract 8 from both sides:
\[
-3x = 2
\]

3. Divide by -3:
\[
x = -\frac{2}{3}
\]

Answer: \( x = -\frac{2}{3} \)

---

#### Problem 3:
\[
\frac{5 - 9x}{6} = -2
\]

1. Multiply both sides by 6 to eliminate the denominator:
\[
5 - 9x = -12
\]

2. Subtract 5 from both sides:
\[
-9x = -17
\]

3. Divide by -9:
\[
x = \frac{17}{9}
\]

Answer: \( x = \frac{17}{9} \)

---

#### Problem 4:
\[
\frac{7x + 6}{3} - 9 = -12
\]

1. Add 9 to both sides:
\[
\frac{7x + 6}{3} = -3
\]

2. Multiply both sides by 3 to eliminate the denominator:
\[
7x + 6 = -9
\]

3. Subtract 6 from both sides:
\[
7x = -15
\]

4. Divide by 7:
\[
x = -\frac{15}{7}
\]

Answer: \( x = -\frac{15}{7} \)

---

Section B

#### Problem 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:
\[
x = -\frac{52}{8} = -\frac{13}{2}
\]

Answer: \( x = -\frac{13}{2} \)

---

#### Problem 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:
\[
x = -5
\]

Answer: \( x = -5 \)

---

#### Problem 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:
\[
x = -\frac{26}{43}
\]

Answer: \( x = -\frac{26}{43} \)

---

Section C

#### Problem 1:
\[
\frac{5x - 2}{3} = \frac{4x + 1}{2}
\]

1. Cross-multiply to eliminate the fractions:
\[
2(5x - 2) = 3(4x + 1)
\]

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:
\[
x = -\frac{7}{2}
\]

Answer: \( x = -\frac{7}{2} \)

---

#### Problem 2:
\[
\frac{7x - 8}{5} = \frac{2x + 5}{4}
\]

1. Cross-multiply to eliminate the fractions:
\[
4(7x - 8) = 5(2x + 5)
\]

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:
\[
x = \frac{57}{18} = \frac{19}{6}
\]

Answer: \( x = \frac{19}{6} \)

---

#### Problem 3:
\[
\frac{-8x - 1}{2} = \frac{5 - 3x}{6}
\]

1. Cross-multiply to eliminate the fractions:
\[
6(-8x - 1) = 2(5 - 3x)
\]

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:
\[
x = -\frac{16}{42} = -\frac{8}{21}
\]

Answer: \( x = -\frac{8}{21} \)

---

Final Answer:

\[
\boxed{x = 14, x = -\frac{2}{3}, x = \frac{17}{9}, x = -\frac{15}{7}, x = -\frac{13}{2}, x = -5, x = -\frac{26}{43}, x = -\frac{7}{2}, x = \frac{19}{6}, x = -\frac{8}{21}}
\]

(Note: This is a partial solution. You can follow the same steps to solve the remaining problems.)