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} & {(7)/(9)c} & {(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} & {(7)/(9)c} & {(2)/(9)r} & (133)/(30) \\ {0w} & {(-31)/(9)c} & {(67)/(9)r} & (328)/(15) \\ {2w} & {9c} & {7r} & {53.70} \\ {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}`
`\begin{bmatrix}{1w} & {(7)/(9)c} & {(2)/(9)r} & (133)/(30) \\ {0w} & {1c} & {0r} & (17137)/(7020) \\ {2w} & {9c} & {7r} & {53.70} \\ {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}`
`\begin{bmatrix}{1w} & {(7)/(9)c} & {(2)/(9)r} & (11449)/(7020) \\ {0w} & {1c} & {0r} & (17137)/(7020) \\ {0w} & {0c} & {1r} & {(28549)/(7020)} \\ {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}`

then

1 bushel of corn is

`C=(17137)/(7020)`
`C=2.44116`

a bushel of corn has a price of $2.441