Great! Let's go through the step-by-step solution for the given equation [tex]\(3(5x + 1) = 13x + 5\)[/tex], filling in the reasons for each step in the provided table format.
\begin{tabular}{|l|l|}
\hline \multicolumn{1}{|c|}{Statement} & \multicolumn{1}{|c|}{Reason} \\
\hline [tex]$3(5x + 1) = 13x + 5$[/tex] & Given equation \\
\hline [tex]$15x + 3 = 13x + 5$[/tex] & Distribute the 3 to terms inside the parentheses \\
\hline [tex]$15x - 13x + 3 = 13x + 5 - 13x$[/tex] & Subtract [tex]$13x$[/tex] from both sides \\
\hline [tex]$2x + 3 = 5$[/tex] & Combine like terms \\
\hline [tex]$2x + 3 - 3 = 5 - 3$[/tex] & Subtract 3 from both sides \\
\hline [tex]$2x = 2$[/tex] & Simplify both sides \\
\hline [tex]$\frac{2x}{2} = \frac{2}{2}$[/tex] & Division property of equality: divide both sides by 2 \\
\hline [tex]$x = 1$[/tex] & Solution simplified \\
\hline
\end{tabular}
To clarify the process:
1. Distribute: Multiply the 3 by both terms inside the parentheses: [tex]\(3(5x + 1) \rightarrow 15x + 3\)[/tex].
2. Combine Like Terms: Move the variable terms to one side and the constants to the other side.
3. Subtract: Isolate the variable by getting rid of the constant term on the same side as the variable.
4. Divide: Solve for the variable by dividing both sides by the coefficient of the variable term.
Thus, the solution to the equation [tex]\(3(5x + 1) = 13x + 5\)[/tex] is [tex]\(x = 1\)[/tex].
