Question

What property shows that these expressions are equivalent?

11
+
6
x
11+6x

8
x
+
(
−
2
x
+
6
)
+
5
8x+(−2x+6)+5

Answer 1

To determine the property that shows the expressions \( 11 + 6x \) and \( 8x + (-2x + 6) + 5 \) are equivalent, we need to simplify the second expression.

Let's simplify \( 8x + (-2x + 6) + 5 \):

1. First, distribute the negative sign in \( -2x + 6 \):
\[
8x + (-2x) + 6 + 5
\]

2. Combine like terms:
\[
8x - 2x + 6 + 5 = (8x - 2x) + (6 + 5) = 6x + 11
\]

Now we have simplified \( 8x + (-2x + 6) + 5 \) down to \( 6x + 11 \).

Next, if we compare \( 11 + 6x \) with \( 6x + 11 \), we can see they are equal since addition is commutative.

Thus, the property that shows that these expressions are equivalent is the **commutative property of addition**, which states that changing the order of the addends does not change the sum. Specifically:

\[
11 + 6x = 6x + 11
\]

This verifies the equivalence of the two expressions.