Question

the steps that a student took to solve the linear equation $4x + 5 = 31 - (x - 4)$ are shown.
the student made an error in one of the steps.
a. in which step does the error first appear?
b. what should be the solution to the original equation?
a. step
b. $x = $

Explanation:

Part A: Find the step with the first error

Step 1: Analyze the Given Equation

The original equation is \( 4x + 5 = 31 - (x - 4) \). To solve this, we first need to simplify the right - hand side by distributing the negative sign. The correct distribution of \( - (x - 4) \) is \( -x + 4 \), so the right - hand side should be \( 31 - x + 4 \).

Step 2: Check Step 1

In Step 1, the student wrote \( 4x + 5 = 31 - x - 4 \). But as we saw from the correct distribution, it should be \( 31 - x + 4 \), not \( 31 - x - 4 \). So the first error occurs in Step 1.

Step 1: Simplify the Right - Hand Side

Given the equation \( 4x+5 = 31-(x - 4) \). Distribute the negative sign on the right - hand side: \( 4x + 5=31 - x + 4 \).

Step 2: Combine Like Terms on the Right - Hand Side

Simplify the right - hand side: \( 4x + 5=35 - x \).

Step 3: Add \( x \) to Both Sides

Add \( x \) to each side of the equation: \( 4x+x + 5=35 - x+x \), which simplifies to \( 5x+5 = 35 \).

Step 4: Subtract 5 from Both Sides

Subtract 5 from both sides: \( 5x+5 - 5=35 - 5 \), so \( 5x = 30 \).

Step 5: Divide Both Sides by 5

Divide both sides by 5: \( \frac{5x}{5}=\frac{30}{5} \), which gives \( x = 6 \).

Answer:

The first error appears in Step 1.

Part B: Solve the original equation