To solve the equation \( 3(x - 5) + 7x = 65 \) step-by-step, we need to apply mathematical properties correctly. Here are the steps and the methods used to justify each one:
1. Starting with the initial equation:
[tex]\[
3(x - 5) + 7x = 65
\][/tex]
2. Distributive Property: Distribute the \( 3 \) across \( (x - 5) \):
[tex]\[
3x - 15 + 7x = 65
\][/tex]
Justification Method: Distributive property.
3. Combine Like Terms: Combine \( 3x \) and \( 7x \):
[tex]\[
10x - 15 = 65
\][/tex]
Justification Method: Combine like terms.
4. Addition Property of Equality: Add \( 15 \) to both sides to isolate the term with \( x \):
[tex]\[
10x = 80
\][/tex]
Justification Method: Addition property of equality.
5. Division Property of Equality: Divide both sides by \( 10 \) to solve for \( x \):
[tex]\[
x = 8
\][/tex]
Justification Method: Division property of equality.
Based on the correctly applied steps and their justifying methods, we can identify the correct table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Solution Step & Method to Justify \\
\hline
[tex]$3x - 15 + 7x = 65$[/tex] & distributive property \\
\hline
[tex]$10x - 15 = 65$[/tex] & combine like terms \\
\hline
[tex]$10x = 80$[/tex] & addition property of equality \\
\hline
[tex]$x = 8$[/tex] & division property of equality \\
\hline
\end{tabular}
\][/tex]
So, the correct table is the second one provided.
