Worksheet for solving algebraic equations with variables on both sides, including an example and six problems.

Problem: Solving Equations with Variables on Both Sides

The task is to solve the given equations by isolating one variable on one side of the equal sign. We will follow the steps outlined in the example:

1. Move variables to one side.
2. Eliminate constants by adding or subtracting.
3. Solve for the variable by multiplying or dividing.

Let's solve each equation step by step.

---

Equation 1: \( n - 3n = 14 - 4n \)

#### Step 1: Simplify both sides
- Left side: \( n - 3n = -2n \)
- Right side: \( 14 - 4n \)

So, the equation becomes:
\[ -2n = 14 - 4n \]

#### Step 2: Move variables to one side
Add \( 4n \) to both sides to eliminate \( -4n \) from the right side:
\[ -2n + 4n = 14 - 4n + 4n \]
\[ 2n = 14 \]

#### Step 3: Solve for \( n \)
Divide both sides by 2:
\[ \frac{2n}{2} = \frac{14}{2} \]
\[ n = 7 \]

#### Solution:
\[ \boxed{n = 7} \]

---

Equation 2: \( 7a + 3 = 4a - 18 \)

#### Step 1: Move variables to one side
Subtract \( 4a \) from both sides to eliminate \( 4a \) from the right side:
\[ 7a - 4a + 3 = 4a - 4a - 18 \]
\[ 3a + 3 = -18 \]

#### Step 2: Eliminate constants
Subtract 3 from both sides to isolate the term with \( a \):
\[ 3a + 3 - 3 = -18 - 3 \]
\[ 3a = -21 \]

#### Step 3: Solve for \( a \)
Divide both sides by 3:
\[ \frac{3a}{3} = \frac{-21}{3} \]
\[ a = -7 \]

#### Solution:
\[ \boxed{a = -7} \]

---

Equation 3: \( 5x - 7 = -10x + 8 \)

#### Step 1: Move variables to one side
Add \( 10x \) to both sides to eliminate \( -10x \) from the right side:
\[ 5x + 10x - 7 = -10x + 10x + 8 \]
\[ 15x - 7 = 8 \]

#### Step 2: Eliminate constants
Add 7 to both sides to isolate the term with \( x \):
\[ 15x - 7 + 7 = 8 + 7 \]
\[ 15x = 15 \]

#### Step 3: Solve for \( x \)
Divide both sides by 15:
\[ \frac{15x}{15} = \frac{15}{15} \]
\[ x = 1 \]

#### Solution:
\[ \boxed{x = 1} \]

---

Equation 4: \( 4 - 7k = 1 - 6k \)

#### Step 1: Move variables to one side
Add \( 6k \) to both sides to eliminate \( -6k \) from the right side:
\[ 4 - 7k + 6k = 1 - 6k + 6k \]
\[ 4 - k = 1 \]

#### Step 2: Eliminate constants
Subtract 4 from both sides to isolate the term with \( k \):
\[ 4 - k - 4 = 1 - 4 \]
\[ -k = -3 \]

#### Step 3: Solve for \( k \)
Multiply both sides by \(-1\) to solve for \( k \):
\[ k = 3 \]

#### Solution:
\[ \boxed{k = 3} \]

---

Equation 5: \( 22 + 4a = 3a - 13 \)

#### Step 1: Move variables to one side
Subtract \( 3a \) from both sides to eliminate \( 3a \) from the right side:
\[ 22 + 4a - 3a = 3a - 3a - 13 \]
\[ 22 + a = -13 \]

#### Step 2: Eliminate constants
Subtract 22 from both sides to isolate the term with \( a \):
\[ 22 + a - 22 = -13 - 22 \]
\[ a = -35 \]

#### Solution:
\[ \boxed{a = -35} \]

---

Equation 6: \( 6y - 9y - 4 = -2y - 2 \)

#### Step 1: Simplify both sides
- Left side: \( 6y - 9y = -3y \), so the equation becomes:
\[ -3y - 4 = -2y - 2 \]

#### Step 2: Move variables to one side
Add \( 2y \) to both sides to eliminate \( -2y \) from the right side:
\[ -3y + 2y - 4 = -2y + 2y - 2 \]
\[ -y - 4 = -2 \]

#### Step 3: Eliminate constants
Add 4 to both sides to isolate the term with \( y \):
\[ -y - 4 + 4 = -2 + 4 \]
\[ -y = 2 \]

#### Step 4: Solve for \( y \)
Multiply both sides by \(-1\) to solve for \( y \):
\[ y = -2 \]

#### Solution:
\[ \boxed{y = -2} \]

---

Final Answers:

1. \( n = 7 \)
2. \( a = -7 \)
3. \( x = 1 \)
4. \( k = 3 \)
5. \( a = -35 \)
6. \( y = -2 \)

\[
\boxed{n = 7, a = -7, x = 1, k = 3, a = -35, y = -2}
\]