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7.2: Explaining Acceptable Moves

Here are some pairs of equations. While one partner listens, the other partner should:

1. Choose a pair of equations from column A. Explain why, if [tex]x[/tex] is a number that makes the first equation true, then it also makes the second equation true.
2. Choose a pair of equations from column B. Explain why the second equation is no longer true for a value of [tex]x[/tex] that makes the first equation true.

Then, switch roles until you run out of time or you run out of pairs of equations.

\begin{tabular}{|c|c|}
\hline & A \\
\hline 1. & \begin{tabular}{l}
[tex]$16 = 4(9 - x)$[/tex] \\
[tex]$16 = 36 - 4x$[/tex]
\end{tabular} \\
\hline 2. & \begin{tabular}{l}
[tex]$5x = 24 + 2x$[/tex] \\
[tex]$3x = 24$[/tex]
\end{tabular} \\
\hline 3. & \begin{tabular}{l}
[tex]$-3(2x + 9) = 12$[/tex] \\
[tex]$2x + 9 = -4$[/tex]
\end{tabular} \\
\hline 4. & \begin{tabular}{l}
[tex]$5x = 3 - x$[/tex] \\
[tex]$5x = -x + 3$[/tex]
\end{tabular} \\
\hline 5. & \begin{tabular}{l}
[tex]$18 = 3x - 6 + x$[/tex] \\
[tex]$18 = 4x - 6$[/tex]
\end{tabular} \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Certainly! Let's explain the solutions for a pair of equations from Column A and Column B with detailed steps.

### Column A

#### Pair 1:
Equations:
1. [tex]\( 16 = 4(9 - x) \)[/tex]
2. [tex]\( 16 = 36 - 4x \)[/tex]

Explanation:

To understand why any [tex]\( x \)[/tex] that makes the first equation true will also make the second equation true, let's manipulate the first equation and show that it can be transformed into the second equation.

1. Start with the first equation:
[tex]\[ 16 = 4(9 - x) \][/tex]

2. Distribute the 4 on the right-hand side:
[tex]\[ 16 = 4 \cdot 9 - 4 \cdot x \][/tex]

3. Perform the multiplication:
[tex]\[ 16 = 36 - 4x \][/tex]

Now, you see that the transformed first equation is identical to the second equation:
[tex]\[ 16 = 36 - 4x \][/tex]

Since we have shown that the first equation can be rearranged into the second equation through distribution and simplification, any value of [tex]\( x \)[/tex] that satisfies the first equation will also satisfy the second equation.

### Column B

#### Pair 1:
Equations:
1. [tex]\( 2x + 5 = 19 \)[/tex]
2. [tex]\( 2x = 14 \)[/tex]

Explanation:

To understand why the second equation is true if the first equation is true, let's simplify the first equation.

1. Start with the first equation:
[tex]\[ 2x + 5 = 19 \][/tex]

2. Subtract 5 from both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[ 2x + 5 - 5 = 19 - 5 \][/tex]

3. Simplify both sides:
[tex]\[ 2x = 14 \][/tex]

Now you see that the simplified form of the first equation is the second equation:
[tex]\[ 2x = 14 \][/tex]

Since we have shown that the first equation can be simplified into the second equation through subtraction, any value of [tex]\( x \)[/tex] that satisfies the first equation will also satisfy the second equation.

### Summary
For both pairs of equations, we showed how the first equation can be manipulated to become the second equation. This manipulation demonstrates that the equations are equivalent and any value of [tex]\( x \)[/tex] that makes one true will also make the other true.

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