Let's fill in the blanks and identify the respective properties used in each step.
Given:
1) [tex]\( 112 \times a = 53 \times 112 \)[/tex]
2) [tex]\((72 \times b) \times 81 = c \times (72 \times 63)\)[/tex]
3) [tex]\((44 \times 25) \times 90 = m \times (25 \times n)\)[/tex]
4) [tex]\((83 \times y) \times 69 = z \times (57 \times 69)\)[/tex]
5) [tex]\(37 \times (56 \times p) = (q \times 71) \times 37\)[/tex]
From these equations, we identified:
1) [tex]\( a = 53 \)[/tex]
2) [tex]\( b = 63, c = 81 \)[/tex]
3) [tex]\( m = 44, n = 90 \)[/tex]
4) [tex]\( y = 57, z = 83 \)[/tex]
5) [tex]\( p = 71, q = 56 \)[/tex]
Now let's fill in the table by providing the variable values and properties used:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
Sentence & Variable Value & Property \\
\hline
1. \( 112 \times a = 53 \times 112 \) & \( a = 53 \) &
\begin{tabular}{l}
Commutative property \\
of multiplication
\end{tabular} \\
\hline
2. \((72 \times b) \times 81 = c \times (72 \times 63)\) & \( b = 63, c = 81 \) &
\begin{tabular}{l}
Associative property \\
of multiplication
\end{tabular} \\
\hline
3. \((44 \times 25) \times 90 = m \times (25 \times n)\) & \( m = 44, n = 90 \) &
\begin{tabular}{l}
Associative property \\
of multiplication
\end{tabular} \\
\hline
4. \((83 \times y) \times 69 = z \times (57 \times 69)\) & \( y = 57, z = 83 \) &
\begin{tabular}{l}
Associative property \\
of multiplication
\end{tabular} \\
\hline
5. \( 37 \times (56 \times p) = (q \times 71) \times 37 \) & \( p = 71, q = 56 \) &
\begin{tabular}{l}
Associative property \\
of multiplication
\end{tabular} \\
\hline
\end{tabular}
\][/tex]
Explanation:
1. In the first equation [tex]\(112 \times a = 53 \times 112\)[/tex], the values of 112 and 53 are interchanged, which is allowed by the commutative property of multiplication.
2. For equations 2, 3, and 4, the grouping of numbers is changed, which is allowed by the associative property of multiplication.
3. Similarly, in equation 5, the associative property is used to change the grouping of the numbers.
Using these properties, we identified the correct values for the variables in each equation, completing the table accurately.
