Question

Hurst's Feed & Seed sold to one customer 5 bushels of wheat, 2 of corn, and 3 of rye, for $31.00. To another customer he sold 2 bushels of wheat, 3 of corn, and 5 of rye, for $27.60. To a third customer he sold 3 bushels of wheat, 5 of corn, and 2 of rye for $32.70. What was the price per bushel for each of the different grains? Let x represent the price per bushel for wheat, y the price per bushel for corn, and z the price per bushel for rye.

Answer 1

Answer:

• wheat: $3.61 per bu
• corn: $3.61 per bu
• rye: $1.91 per bu

Step-by-step explanation:

You want to know the price per bushel for wheat, corn, and rye, given the sales ...

• 5 bu wheat + 2 bu corn + 3 bu rye for $31.00
• 2 bu wheat + 3 bu corn + 5 bu rye for $27.60
• 3 bu wheat + 5 bu corn + 2 bu rye for $32.70

Equations

Using x, y, and z for price per bushel of wheat, corn, and rye, respectively, the given sales tell us ...

• 5x +2y +3z = 31.00
• 2x +3y +5z = 27.60
• 3x +5y +2z = 32.70

Solution

The easiest way to solve a system of equations like this is to make use of the matrix manipulation capabilities of a calculator. The attached calculator display shows the reduction of the augmented matrix of coefficients to "reduced row-echelon form". That puts the solution in the last column:

  x = y = 3.61, z = 1.91

The price per bushel for wheat and corn is the same: $3.61; the price per bushel for rye is $1.91.

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Additional comment

The second attachment shows a solution using a graphing calculator. In this solution, an expression for z based on the first equation is used in the remaining two equations, which then tell us the values of x and y.

If you add all of the equations, you get 10x +10y +10z = 91.30, which gives you ...

  x + y + z = 9.13

This can be solved for any variable you may want to substitute into any pair of the equations. This approach avoids any messy fractions.

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