Question

Problem Solving with Multi-Step Equation

Answer 1

Sure! Here’s a step-by-step approach to solving multi-step equations, along with an example to illustrate the process.

### Steps to Solve a Multi-Step Equation:

1. **Simplify Both Sides**: Distribute any factors and combine like terms on both sides of the equation.

2. **Move Variable Terms to One Side**: Use addition or subtraction to get all terms with the variable on one side, and constant terms on the other.

3. **Isolate the Variable**: Once the variable terms are on one side, use division or multiplication to isolate the variable.

4. **Check Your Solution**: Substitute your solution back into the original equation to ensure that both sides are equal.

### Example Problem:

Let's solve the equation:
\[ 3(x + 4) - 5 = 2(x - 1) + 9 \]

#### Step 1: Simplify Both Sides
Apply the distributive property:
\[ 3x + 12 - 5 = 2x - 2 + 9 \]

Combine like terms:
\[ 3x + 7 = 2x + 7 \]

#### Step 2: Move Variable Terms to One Side
Subtract \(2x\) from both sides:
\[ 3x - 2x + 7 = 7 \]

This simplifies to:
\[ x + 7 = 7 \]

#### Step 3: Isolate the Variable
Subtract 7 from both sides:
\[ x = 0 \]

#### Step 4: Check Your Solution
Substitute \(x = 0\) back into the original equation:
\[ 3(0 + 4) - 5 = 2(0 - 1) + 9 \]
This simplifies to:
\[ 12 - 5 = -2 + 9 \]
\[ 7 = 7 \]

Since both sides of the equation are equal, our solution \(x = 0\) is correct.

### Additional Example:

Let’s try another equation:
\[ 5(x - 2) + 3 = 2x + 15 \]

**Step 1: Simplify Both Sides**
Distribute the \(5\):
\[ 5x - 10 + 3 = 2x + 15 \]
Combine like terms:
\[ 5x - 7 = 2x + 15 \]

**Step 2: Move Variable Terms to One Side**
Subtract \(2x\) from both sides:
\[ 5x - 2x - 7 = 15 \]
This simplifies to:
\[ 3x - 7 = 15 \]

**Step 3: Isolate the Variable**
Add \(7\) to both sides:
\[ 3x = 22 \]
Now, divide by \(3\):
\[ x = \frac{22}{3} \]

**Step 4: Check Your Solution**
Substituting \(x = \frac{22}{3}\) back into the original equation:
\[ 5\left(\frac{22}{3} - 2\right) + 3 = 2\left(\frac{22}{3}\right) + 15 \]
Calculating:
\[ 5\left(\frac{22}{3} - \frac{6}{3}\right) + 3 = 2\left(\frac{22}{3}\right) + 15 \]
\[ 5\left(\frac{16}{3}\right) + 3 = \frac{44}{3} + 15 \]
Multiply:
\[ \frac{80}{3} + \frac{9}{3} = \frac{44}{3} + \frac{45}{3} \]
Combine:
\[ \frac{89}{3} = \frac{89}{3} \]

Both sides equal, so \(x = \frac{22}{3}\) is indeed the correct solution.

Feel free to reach out if you have any specific types of equations you'd like help with!