Find equivalent expressions for 7x + 35 and xy + 5y + 2x + 10. Explain your steps.

To find an equivalent expression for $7x + 35$, Lucy needs to find the Highest Common Factor (HCF) of the two terms, $7x$ and 35.
The HCF of the numerical coefficients (7 and 35) is 7.
There is no common variable factor.
So, the HCF of the expression is 7.
Then, divide each term by the HCF and write the expression as the HCF multiplied by the sum of the results.

The equivalent expression is $\boxed{7(x + 5)}$.

There is no common factor (other than 1) that divides all four terms ($xy$, $5y$, $2x$, 10).

In Group 1 ($xy + 5y$), the common factor is $y$.
In Group 2 ($2x + 10$), the common factor is 2.

Factor Group 1: $xy + 5y = y(x + 5)$
Factor Group 2: $2x + 10 = 2(x + 5)$
Now, substitute these back into the grouped expression:

Notice that $(x + 5)$ is a common factor in both terms. Factor it out:

The equivalent algebraic expression is $\boxed{(x + 5)(y + 2)}$.