Let's break down each of the steps in your solution to the equation \( 24(34 + x) = 1,200 \) and explain what was done to arrive at each step, along with the corresponding reasoning from the provided answers.
Step 1: \( 816 + 24x = 1,200 \)
Reasoning: Distributive property
Explanation: In this step, we applied the distributive property to expand the left side of the equation. Specifically, we multiplied \( 24 \) by both \( 34 \) and \( x \):
\[
24(34) + 24(x) = 1,200 \implies 816 + 24x = 1,200.
\]
Step 2: \( 816 - 816 + 24x = 1,200 - 816 \)
Reasoning: Subtract as the inverse operation to isolate the variable
Explanation: Here, we subtract \( 816 \) from both sides of the equation to move the constant term to the right side. This helps us to isolate the term containing \( x \):
\[
816 - 816 + 24x = 1,200 - 816 \implies 0 + 24x = 384.
\]
Step 3: \( 24x = 384 \)
Reasoning: Combine like terms
Explanation: After the subtraction in Step 2, we simplify the equation. Since \( 0 + 24x \) is just \( 24x \), we have:
\[
24x = 384.
\]
Step 4: \( x = 16 \)
Reasoning: Division as the inverse operation to isolate the variable
Explanation: Finally, to solve for \( x \), we divide both sides of the equation by \( 24 \) to isolate \( x \):
\[
x = \frac{384}{24} \implies x = 16.
\]
Summary
- Step 1: Distributive property
- Step 2: Subtract as the inverse operation to isolate the variable
- Step 3: Combine like terms
- Step 4: Division as the inverse operation to isolate the variable
This step-by-step breakdown illustrates the methods used at each stage in solving the equation and aligns reasoning with the specific operations performed.