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!