Alright, let's evaluate each expression and determine which expressions are equivalent.
First, let's look at the expression:
[tex]\[ 5x + 10 \][/tex]
We need to identify whether any of the following expressions are equivalent:
- [tex]\( 5(x + 2) \)[/tex]
- [tex]\( 5 \cdot x + 5 \cdot 2 \)[/tex]
- [tex]\( 15x \)[/tex]
- [tex]\( 5(2x + 1) \)[/tex]
### Evaluation of Expressions for [tex]\(5x + 10\)[/tex]:
1. Expression [tex]\(5(x + 2)\)[/tex]:
Applying the distributive property:
[tex]\[
5(x + 2) = 5 \cdot x + 5 \cdot 2 = 5x + 10
\][/tex]
This is equivalent to [tex]\(5x + 10\)[/tex].
2. Expression [tex]\(5 \cdot x + 5 \cdot 2\)[/tex]:
Simplifying:
[tex]\[
5 \cdot x + 5 \cdot 2 = 5x + 10
\][/tex]
This is also equivalent to [tex]\(5x + 10\)[/tex].
3. Expression [tex]\(15x\)[/tex]:
This expression is already in its simplest form and does not simplify to [tex]\(5x + 10\)[/tex].
4. Expression [tex]\(5(2x + 1)\)[/tex]:
Applying the distributive property:
[tex]\[
5(2x + 1) = 5 \cdot 2x + 5 \cdot 1 = 10x + 5
\][/tex]
This is not equivalent to [tex]\(5x + 10\)[/tex].
Based on the evaluations, the equivalent expressions to [tex]\(5x + 10\)[/tex] are:
- [tex]\( 5(x + 2) \)[/tex]
- [tex]\( 5 \cdot x + 5 \cdot 2 \)[/tex]
Next, let’s consider the expression:
[tex]\[ 15 + 16y - 7 - y \][/tex]
We need to identify whether any of the following expressions are equivalent:
- [tex]\( 8y + 15y \)[/tex]
- [tex]\( 8y + 15 \)[/tex]
- [tex]\( 15 + 8y \)[/tex]
- [tex]\( 15y + 8 \)[/tex]
### Evaluation of Expressions for [tex]\(15 + 16y - 7 - y\)[/tex]:
First, simplify the original expression:
[tex]\[ 15 + 16y - 7 - y = 15 + 15y - 7 \][/tex]
Combine like terms:
[tex]\[ 15 - 7 + 15y = 8 + 15y \][/tex]
1. Expression [tex]\(8y + 15y\)[/tex]:
Simplifying:
[tex]\[
8y + 15y = 23y
\][/tex]
This is not equivalent to [tex]\(8 + 15y\)[/tex].
2. Expression [tex]\(8y + 15\)[/tex]:
This remains as [tex]\(8y + 15\)[/tex], which is not equivalent to [tex]\(8 + 15y\)[/tex].
3. Expression [tex]\(15 + 8y\)[/tex]:
Rearrange:
[tex]\[
15 + 8y = 8y + 15
\][/tex]
This is equivalent to [tex]\(8 + 15y\)[/tex].
4. Expression [tex]\(15y + 8\)[/tex]:
This is already in its simplest form and is equivalent to [tex]\(8 + 15y\)[/tex].
Based on the evaluations, the equivalent expressions to [tex]\(15 + 16y - 7 - y\)[/tex] are:
- [tex]\(15 + 8y\)[/tex]
- [tex]\(15y + 8\)[/tex]
To summarize:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
& 5(x+2) & 5 \cdot x + 5 \cdot 2 & 15x & 5(2x+1) \\
\hline
5x + 10 & \checkmark & \checkmark & & \\
\hline
(b) & 8y + 15y & 8y + 15 & 15 + 8y & 15y + 8 \\
\hline
15 + 16y - 7 - y & & & \checkmark & \checkmark \\
\hline
\end{array}
\][/tex]
