Graphing Linear Equations Worksheets - Free Printable

The image you provided is a worksheet titled "Completing Linear Equations," which involves solving linear equations using the distributive property and combining like terms. Below, I will explain how to solve each problem step by step.

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Problem 1:

$$
3(x + 4) = 27
$$

#### Step-by-Step Solution:
1. Distribute the 3 across the parentheses:
$$
3(x + 4) = 3 \cdot x + 3 \cdot 4 = 3x + 12
$$
So the equation becomes:
$$
3x + 12 = 27
$$

2. Isolate the term with \( x \):
Subtract 12 from both sides of the equation:
$$
3x + 12 - 12 = 27 - 12
$$
Simplify:
$$
3x = 15
$$

3. Solve for \( x \):
Divide both sides by 3:
$$
\frac{3x}{3} = \frac{15}{3}
$$
Simplify:
$$
x = 5
$$

#### Final Answer:
$$
\boxed{x = 5}
$$

---

Problem 2:

$$
2(3y - 5) = 14
$$

#### Step-by-Step Solution:
1. Distribute the 2 across the parentheses:
$$
2(3y - 5) = 2 \cdot 3y - 2 \cdot 5 = 6y - 10
$$
So the equation becomes:
$$
6y - 10 = 14
$$

2. Isolate the term with \( y \):
Add 10 to both sides of the equation:
$$
6y - 10 + 10 = 14 + 10
$$
Simplify:
$$
6y = 24
$$

3. Solve for \( y \):
Divide both sides by 6:
$$
\frac{6y}{6} = \frac{24}{6}
$$
Simplify:
$$
y = 4
$$

#### Final Answer:
$$
\boxed{y = 4}
$$

---

Problem 3:

$$
4(2z + 3) - 8 = 20
$$

#### Step-by-Step Solution:
1. Distribute the 4 across the parentheses:
$$
4(2z + 3) = 4 \cdot 2z + 4 \cdot 3 = 8z + 12
$$
So the equation becomes:
$$
8z + 12 - 8 = 20
$$

2. Combine like terms:
Simplify the left side:
$$
8z + 12 - 8 = 8z + 4
$$
So the equation becomes:
$$
8z + 4 = 20
$$

3. Isolate the term with \( z \):
Subtract 4 from both sides of the equation:
$$
8z + 4 - 4 = 20 - 4
$$
Simplify:
$$
8z = 16
$$

4. Solve for \( z \):
Divide both sides by 8:
$$
\frac{8z}{8} = \frac{16}{8}
$$
Simplify:
$$
z = 2
$$

#### Final Answer:
$$
\boxed{z = 2}
$$

---

Problem 4:

$$
5(2a - 1) + 3 = 38
$$

#### Step-by-Step Solution:
1. Distribute the 5 across the parentheses:
$$
5(2a - 1) = 5 \cdot 2a - 5 \cdot 1 = 10a - 5
$$
So the equation becomes:
$$
10a - 5 + 3 = 38
$$

2. Combine like terms:
Simplify the left side:
$$
10a - 5 + 3 = 10a - 2
$$
So the equation becomes:
$$
10a - 2 = 38
$$

3. Isolate the term with \( a \):
Add 2 to both sides of the equation:
$$
10a - 2 + 2 = 38 + 2
$$
Simplify:
$$
10a = 40
$$

4. Solve for \( a \):
Divide both sides by 10:
$$
\frac{10a}{10} = \frac{40}{10}
$$
Simplify:
$$
a = 4
$$

#### Final Answer:
$$
\boxed{a = 4}
$$

---

Problem 5:

$$
6(3b + 2) - 12 = 42
$$

#### Step-by-Step Solution:
1. Distribute the 6 across the parentheses:
$$
6(3b + 2) = 6 \cdot 3b + 6 \cdot 2 = 18b + 12
$$
So the equation becomes:
$$
18b + 12 - 12 = 42
$$

2. Combine like terms:
Simplify the left side:
$$
18b + 12 - 12 = 18b
$$
So the equation becomes:
$$
18b = 42
$$

3. Solve for \( b \):
Divide both sides by 18:
$$
\frac{18b}{18} = \frac{42}{18}
$$
Simplify:
$$
b = \frac{7}{3}
$$

#### Final Answer:
$$
\boxed{b = \frac{7}{3}}
$$

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Summary of Answers:

1. \( x = 5 \)
2. \( y = 4 \)
3. \( z = 2 \)
4. \( a = 4 \)
5. \( b = \frac{7}{3} \)

$$
\boxed{x = 5, y = 4, z = 2, a = 4, b = \frac{7}{3}}
$$