Question

A grain dealer sold to one customer 9 bushels of wheat, 7 of corn, and 2 of rye, for $39.90; to another, 7 of wheat, 2 of corn, and 9 of rye, for $52.20; and to a third, 2 of wheat, 9 of corn, and 7 of rye, for $53.70. What was the price per bushel for corn?

Answer 1

this is a 3 system equation

we have 3 equation and 3 variables

$\begin{bmatrix}{9w} & {7c} & {2r} & {39.90} \\ {7w} & {2c} & {9r} & {52.20} \\ {2w} & {9c} & {7r} & {53.70} \\ {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}$

where

w=wheat

c=corn

r=rye

price per bushell of corn, c=?

Gauss-Jordan method

$\begin{bmatrix}{1w} & {\frac{7}{9}c} & {\frac{2}{9}r} & \frac{{39.90}}{9} \\ {7w} & {2c} & {9r} & {52.20} \\ {2w} & {9c} & {7r} & {53.70} \\ {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}$$\begin{bmatrix}{1w} & {\frac{7}{9}c} & {\frac{2}{9}r} & \frac{133}{30} \\ {0w} & {\frac{-31}{9}c} & {\frac{67}{9}r} & \frac{328}{15} \\ {2w} & {9c} & {7r} & {53.70} \\ {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}$$\begin{bmatrix}{1w} & {\frac{7}{9}c} & {\frac{2}{9}r} & \frac{133}{30} \\ {0w} & {1c} & {0r} & \frac{17137}{7020} \\ {2w} & {9c} & {7r} & {53.70} \\ {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}$$\begin{bmatrix}{1w} & {\frac{7}{9}c} & {\frac{2}{9}r} & \frac{11449}{7020} \\ {0w} & {1c} & {0r} & \frac{17137}{7020} \\ {0w} & {0c} & {1r} & {\frac{28549}{7020}} \\ {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}$

then

1 bushel of corn is

$C=\frac{17137}{7020}$$C=2.44116$

a bushel of corn has a price of $2.441