Defining and expressing as a system of two equations. Is my answer good?

Question

We wish to spend $\$164.00$ by purchasing $10$ books, some costing $\$15.00$ and other $\$17.00$. How many books of each price do we buy?

My answer: let $x$ = number of books costing $\$15.00$ and let $y$ = number of books costing $\$17.00$.

For my first equation I'll say, $x + y = 10$ and my second equation will be $15x + 17y = 164$.

Answer 1

Yes, it's right. Translating your equations into English: $x+y=10$ means that the total number of books is $10$, and $15x+17y$ means that the total amount spent is $\$164.00$. Hence, yes. Indeed, the answer is $y=7,x=3$.

Answer 2

You set up the equations right. Multiplying both sides of your first equation by $15$ yields $15x+15y = 150$. Now, if we subtract this from your second equation, we get that $2y = 14$, so $y = 7$. $10-7 = 3$, so $\boxed{(x,y) = (3,7)}$.