A farmer harvested $1$ section (which is $640\, acres$) of wheat and $2$ sections of barley. The total yield of grain for both areas was $99,840\, bushels$. The wheat sold for $6.35\, /bushel$ The barley sold for $2.70\, / bushel$. The farmer received $363,008$ for both crops. What was the yield of each section in $bushel\, /acres$
Here is what I know. I need to make a system of equation so I let $x$ stand for wheat and $y$ stand for barley. There is $1$ section of wheat which is $640\, acres$ and 2 sections of barley which is $1280\, acres$. Therefore, the two equations become: \begin{align*} &x+2y=99840 \\ &6.35x+2.70y=363008 \end{align*} To find the value of $y$ in the second equation I move $2y$ to the right side of the first equation and continue as follow:
\begin{align*} &x=99840-2y\\ &6.35(99840-2y)+2.70y=363008\\ &633984-12.7y+2.7y=363008\\ &-10y=363008-633984\\ &-10y=-270976\\ &y=27097.6\\ \end{align*} The total of barley is $27097.6\, bushels$ therefore dividing that by $1280\, acres$ I get $21.17 \, bushels\, /acres$
Replacing $y$ with $27097.6$ in the first equation I get $x=45644.8$ total wheat, dividing it by $640\, acres$ gives me $71.32\, bushels/acres$
I verified the solution by substituting $x$ and $y$ with the numbers in both equations and the solution is verified. However the book answers are $40\, bushels/acres$ of wheat and $58 \, bushels/acres$ of barley.
The book does not give any explanation or show the steps it took. I finished high school 20 years ago and I am studying on my own preparing for university therefore I have no access to a teacher. So I appreciate it if someone can explain where I went wrong and the steps I need to take.
The equation first is wrong its simply $x+y=99840$. And second equation is correct . So multiply first with $6.35$ you get $6.35x+6.35y=633984$. Subtracting with equation 2 ie$6.35x+2.70y=363008$ you get $y=74240$ we want $bushels/area$ so $\frac{74240}{1280}=58$ similarly plugging value of 'y' you get $x$ which is $40$. Here we take section of barley as $y$ as we dont know whether 2 sections are evenly distributed or not. Hope its clear.