Let's analyze both expressions step-by-step and identify the equivalent expressions.
### Part (a): [tex]\( 4(5x - 2) \)[/tex]
First, we'll expand and simplify the expression [tex]\( 4(5x - 2) \)[/tex]:
[tex]\[ 4(5x - 2) = 4 \cdot 5x - 4 \cdot 2 \][/tex]
[tex]\[ = 20x - 8 \][/tex]
Next, let's compare this with the given options:
1. [tex]\( 20x - 2 \)[/tex]: This is not equivalent to [tex]\( 20x - 8 \)[/tex].
2. [tex]\( 20x - 8 \)[/tex]: This is the same as our simplified expression.
3. [tex]\( 9x - 2 \)[/tex]: This is not equivalent to [tex]\( 20x - 8 \)[/tex].
4. [tex]\( 4 \cdot 5x - 4 \cdot 2 \)[/tex]: This matches our original way of writing the expansion and is equivalent.
So, the equivalent expressions for part (a) are:
- [tex]\( 20x - 8 \)[/tex]
- [tex]\( 4 \cdot 5x - 4 \cdot 2 \)[/tex]
### Part (b): [tex]\( 10y + 5y + 8y \)[/tex]
Now, let's simplify the expression [tex]\( 10y + 5y + 8y \)[/tex]:
[tex]\[ 10y + 5y + 8y = (10 + 5 + 8)y \][/tex]
[tex]\[ = 23y \][/tex]
Next, let's compare this with the given options:
1. [tex]\( 23y \)[/tex]: This is the same as our simplified expression.
2. [tex]\( 13y - 10y \)[/tex]: This is equivalent to [tex]\( 3y \)[/tex], not [tex]\( 23y \)[/tex].
3. [tex]\( 10y + 13y \)[/tex]: This is equivalent to [tex]\( (10 + 13)y = 23y \)[/tex].
4. [tex]\( 23 + y \)[/tex]: This does not simplify to [tex]\( 23y \)[/tex].
So, the equivalent expressions for part (b) are:
- [tex]\( 23y \)[/tex]
- [tex]\( 10y + 13y \)[/tex]
### Summary:
The equivalent expressions are:
- For [tex]\( 4(5x - 2) \)[/tex]: [tex]\( 20x - 8 \)[/tex], [tex]\( 4 \cdot 5x - 4 \cdot 2 \)[/tex]
- For [tex]\( 10y + 5y + 8y \)[/tex]: [tex]\( 23y \)[/tex], [tex]\( 10y + 13y \)[/tex]
