Solve.
-A grain dealer sold to one customer 5 bushels of wheat, 2 of corn, and 3 of rye, for $21.50; to another, 2 of wheat, 3 of corn, and 5 of rye, for $19.00; and to a third, 3 of wheat, 5 of corn, and 2 of rye, for $26.50. What was the
Price per bushel for corn?

Question

Quizplus · Accepted Answer

Let's denote the price per bushel of wheat, corn, and rye by w, c, and r respectively. Then we can write the following system of equations:
5w + 2c + 3r = 21.5
2w + 3c + 5r = 19
3w + 5c + 2r = 26.5
We need to solve for c, the price per bushel of corn. One way to do this is to use elimination method. Let's multiply the second equation by 3 and subtract it from the third equation multiplied by 2:
(2w + 3c + 5r) x 2 - (3w + 5c + 2r) x 3 = 0
Simplifying this gives:
-w + 7c - 11r = -16.5
Now let's multiply the first equation by 3 and subtract it from the second equation multiplied by 5:
(5w + 2c + 3r) x 5 - (2w + 3c + 5r) x 3 = 0
Simplifying this gives:
13w - 11c - 7r = 34.5
Finally, let's multiply the first equation by 2 and subtract it from the third equation multiplied by 5:
(3w + 5c + 2r) x 5 - (5w + 2c + 3r) x 2 = 0
Simplifying this gives:
7w + 16c - 11r = 29.5
Now we have a system of three equations in three variables. We can solve for w, c, and r using any method of our choice (substitution, elimination, matrix inversion, etc). For example, using elimination method again, we can eliminate w by adding the last two equations:
(13w - 11c - 7r) + (7w + 16c - 11r) = 63.5
Simplifying this gives:
20c - 18r = 17
Now we can solve for c by isolating it in terms of r:
20c = 18r + 17
c = (9/10)r + 17/20
Substituting this expression for c into any of the equations involving w and r will give us the values of w and r, and therefore the value of c. After some algebraic manipulation, we can find that:
c = 354/100 = 3.54
Therefore, the price per bushel of corn is $3.54.
The best choice is B, since the solution we obtained is unique and satisfies all the given constraints. Let x, y, and z be the prices per bushel for wheat, corn, and rye respectively. 
Then we have the following system of equations:
5x + 2y + 3z = 21.50
2x + 3y + 5z = 19.00
3x + 5y + 2z = 26.50

Solving this system of equations using any method of your choice, we get y = $3.54 per bushel of corn. Therefore, the answer is choice B.

Solve. -A Grain Dealer Sold to One Customer 5 Bushels of of Wheat