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!