Chapter 5.2, Problem 19E

To determine

To Write: A system of linear equation that represents the situation “A farmer plants corn and wheat on a 180 acre farm. The farmer wants to plant three times as many acres of corn as wheat”, and to find how many acres of each crop should the farmer plant.

Answer to Problem 19E

The required system of linear equation is $\begin{array}{l}x+y=180\\ y=3x\end{array}$

And the farmer should plant 45 acres of corn and 135 acres of wheat.

Explanation of Solution

Given information:

A farmer plants corn and wheat on a 180 acre farm.

The farmer wants to plant three times as many acres of corn as wheat.

Calculation:

Let the acres of corn plant be “x”

Let the acres of wheat plant be “y”

Since the farmer plants corn and wheat on a 180 acre farm,

The first equation is $x+y=180$

Since the farmer wants to plant three times as many of corn as wheat

Therefore, the second equation is $y=3x$

Therefore yhe required system of linear equation is : $\begin{array}{l}x+y=180\\ y=3x\end{array}$

Equation 2 is already solved for y

  $y=3x$

Substitute 3x for y in equation 1 and solve for x

  $\begin{array}{l}x+y-180\\ x+3x=180\\ 4x=180\end{array}$

(Simplify)

  $\frac{4x}{4}=\frac{180}{4}$

(Divide each side by 4)

  $x=45$

(Simplify)

Now, substitute 45 for x in equation 2 and solve for y.

  $\begin{array}{l}y=3x\\ y=3\left(45\right)\\ y=135\end{array}$

(Simplify)

Therefore, the required solution is $\left(45,135\right)$

i.e. The farmer should plant 45 acres of corn and 135 acres of wheat.