Question

Simplify the sum. State any restrictions on the variable.
(8y+8)/(y-2)+(7y+9)/y-2)
Select the correct choice below and fill in the answer box(es) to complete your choice.
1(8y+8)/(y-2)+(7y+9)/(y-2)=____ for all used values of y (simplify your answer).
2)(8y+8)/(y-2)/(7y+9)/(y-2)=____ for y(=/)____ (Simplify your answers. Use a comma to separate answers as needed.)

Answer 1

Let's simplify the expression step by step. The expression given is:

\[
\frac{8y + 8}{y - 2} + \frac{7y + 9}{y - 2}
\]

Since both fractions have the same denominator, we can combine them:

\[
\frac{(8y + 8) + (7y + 9)}{y - 2}
\]

Now, simplify the numerator:

\[
8y + 8 + 7y + 9 = 15y + 17
\]

So, the expression becomes:

\[
\frac{15y + 17}{y - 2}
\]

Next, we need to state any restrictions on the variable \( y \). The only restriction comes from the denominator, which cannot be equal to zero:

\[
y - 2 \neq 0 \implies y \neq 2
\]

Thus, our simplified expression is:

\[
\frac{15y + 17}{y - 2} \quad \text{for all } y \neq 2.
\]

Now regarding the second part of the question, we see that you might have meant to ask about products or division, but the given equation had a + sign, not a division.

Assuming you want to simplify the following expression as well:

\[
\frac{\frac{8y + 8}{y - 2}}{\frac{7y + 9}{y - 2}}
\]

This can also be simplified. When you divide two fractions, you multiply by the reciprocal:

\[
\frac{8y + 8}{y - 2} \div \frac{7y + 9}{y - 2} = \frac{8y + 8}{y - 2} \cdot \frac{y - 2}{7y + 9}
\]

The \(y - 2\) in the numerator and denominator cancel out (as long it does not equal zero):

\[
\frac{8y + 8}{7y + 9}
\]

Now we set restrictions for this case again. The restriction remains \(y \neq 2\). Additionally, we should check the denominator \(7y + 9\):

\[
7y + 9 \neq 0 \implies y \neq -\frac{9}{7}
\]

Thus, the final answer for the second expression is:

\[
\frac{8y + 8}{7y + 9} \quad \text{for } y \neq 2, -\frac{9}{7}
\]

To summarize:

1) \(\frac{8y + 8}{y - 2} + \frac{7y + 9}{y - 2} = \frac{15y + 17}{y - 2} \text{ for all } y \neq 2\)

2) \(\frac{\frac{8y+8}{y-2}}{\frac{7y+9}{y-2}} = \frac{8y+8}{7y+9} \text{ for } y \neq 2, -\frac{9}{7}\)