Sure, let's go through the mathematical solution for filling in the missing justifications step by step.
[tex]\[
\begin{tabular}{|l|l|}
\hline Mathematical Statement & \multicolumn{1}{|c|}{ Justification } \\
\hline $4x + 3 = x + 5 - 2x$ & Given \\
\hline $4x + 3 = x - 2x + 5$ & Commutative Property of Addition \\
\hline
\end{tabular}
\][/tex]
Step 1: Combine Like Terms
First, we combine like terms on the right side of the equation:
[tex]\[
4x + 3 = -x + 5
\][/tex]
[tex]\[
\begin{tabular}{|l|l|}
\hline $4x + 3 = x - 2x + 5$ & Commutative Property of Addition \\
\hline $4x + 3 = -x + 5$ & Combine Like Terms \\
\hline
\end{tabular}
\][/tex]
Step 2: Addition Property of Equality
Next, we want to simplify the equation further by isolating terms involving [tex]\(x\)[/tex] on one side. We add [tex]\( x \)[/tex] to both sides:
[tex]\[
4x + 3 + x = -x + 5 + x
\][/tex]
[tex]\[
5x + 3 = 5
\][/tex]
[tex]\[
\begin{tabular}{|l|l|}
\hline $4x + 3 = -x + 5$ & Combine Like Terms \\
\hline $5x + 3 = 5$ & Addition Property of Equality \\
\hline
\end{tabular}
\][/tex]
Step 3: Subtraction Property of Equality
Now, we need to isolate the term involving [tex]\( x \)[/tex] by removing the constant term from the left side. We subtract 3 from both sides:
[tex]\[
5x + 3 - 3 = 5 - 3
\][/tex]
[tex]\[
5x = 2
\][/tex]
[tex]\[
\begin{tabular}{|l|l|}
\hline $5x + 3 = 5$ & Addition Property of Equality \\
\hline $5x = 2$ & Subtraction Property of Equality \\
\hline
\end{tabular}
\][/tex]
Step 4: Division Property of Equality
Finally, to solve for [tex]\( x \)[/tex], we divide both sides by 5:
[tex]\[
x = \frac{2}{5}
\][/tex]
[tex]\[
\begin{tabular}{|l|l|}
\hline $5x = 2$ & Subtraction Property of Equality \\
\hline $x = \frac{2}{5}$ & Division Property of Equality \\
\hline
\end{tabular}
\][/tex]
So, the correctly filled-in justifications in the correct order are:
- Combine Like Terms
- Addition Property of Equality
- Subtraction Property of Equality
- Division Property of Equality
