Question

When solving an equation, Carmen's first step is shown below. Which property justifies Carmen's first step?
Original Equation:
Original Equation:
21, left parenthesis, x, squared, plus, 2, right parenthesis, left parenthesis, one third, right parenthesis, minus, 2, equals, minus, 9, x, squared, plus, 2
21(x
2
+2)(
3
1
​
)−2=
−9x
2
+2
First Step:
First Step:
7, left parenthesis, x, squared, plus, 2, right parenthesis, minus, 2, equals, minus, 9, x, squared, plus, 2
7(x
2
+2)−2=
−9x
2
+2

Answer 1

Carmen's first step involves simplifying the equation by distributing the constant \(21\) and the fraction \(\frac{1}{3}\). The property that justifies this step is the **Associative Property of Multiplication**.

Here’s a breakdown:

In the original equation:

\[
21 \left( x^2 + 2 \right) \left( \frac{1}{3} \right) - 2 = -9x^2 + 2
\]

Carmen simplifies \(21 \cdot \frac{1}{3}\) to \(7\):

\[
21 \cdot \frac{1}{3} = 7
\]

Thus, the equation simplifies to:

\[
7 \left( x^2 + 2 \right) - 2 = -9x^2 + 2
\]

This step is a result of applying the **Associative Property of Multiplication**, which allows you to rearrange and group factors in multiplication.