Question

Problem solving with multi-step equations

Answer 1

Certainly! Solving multi-step equations involves a series of steps that simplify the equation to isolate the variable (usually represented as \( x \)). Here are some fundamental steps you can follow, along with example problems.

### Steps for Solving Multi-Step Equations

1. **Simplify Both Sides**: If there are parentheses, distribute as necessary. Combine like terms on both sides if possible.

2. **Move the Variable**: Use addition or subtraction to get all terms containing the variable on one side of the equation and constant terms on the other.

3. **Isolate the Variable**: Once you've moved the variables to one side and constants to the other, use division or multiplication to solve for the variable.

4. **Check Your Solution**: Plug the solution back into the original equation to verify that it works.

### Example Problems

#### Example 1:
Solve the equation:
\[ 2(x + 3) = 16 \]

**Step 1**: Distribute on the left side:
\[ 2x + 6 = 16 \]

**Step 2**: Subtract 6 from both sides:
\[ 2x = 10 \]

**Step 3**: Divide by 2:
\[ x = 5 \]

**Check**:
Plug \( x = 5 \) into the original equation:
\[ 2(5 + 3) = 16 \]
\[ 2(8) = 16 \] - True.

#### Example 2:
Solve the equation:
\[ 3x - 4 = 2x + 5 \]

**Step 1**: Move \( 2x \) to the left side:
\[ 3x - 2x - 4 = 5 \]

**Step 2**: Combine like terms:
\[ x - 4 = 5 \]

**Step 3**: Add 4 to both sides:
\[ x = 9 \]

**Check**:
Plug \( x = 9 \) into the original equation:
\[ 3(9) - 4 = 2(9) + 5 \]
\[ 27 - 4 = 18 + 5 \]
\[ 23 = 23 \] - True.

#### Example 3:
Solve the equation:
\[ 4(2x - 1) = 3(x + 5) + 2 \]

**Step 1**: Distribute on both sides:
\[ 8x - 4 = 3x + 15 + 2 \]
\[ 8x - 4 = 3x + 17 \]

**Step 2**: Move \( 3x \) to the left side:
\[ 8x - 3x - 4 = 17 \]

**Step 3**: Combine like terms:
\[ 5x - 4 = 17 \]

**Step 4**: Add 4 to both sides:
\[ 5x = 21 \]

**Step 5**: Divide by 5:
\[ x = \frac{21}{5} \] or \( x = 4.2 \)

**Check**:
Plug \( x = \frac{21}{5} \) back into the original equation to verify.

### Practice Problems
1. Solve for \( x \): \( 5(x - 2) + 6 = 3x + 12 \)
2. Solve for \( x \): \( 7x + 1 - 4 = 5x + 3 \)
3. Solve for \( x \): \( 8(2x + 1) = 4(x + 7) \)

Feel free to ask for explanations, clarifications, or additional problems!