Chapter 6 - Systems of Equations and Inequalities - 6-2 Solving Systems Using Substitution - Practice and Problem-Solving Exercises - Page 373: 42

$110$ acres of corn $70$ acres of tomatoes $140$ acres of sunflowers

Let the number of acres of corn be $x$ , the number of acres of tomatoes be $y$ , and the number of acres of sunflowers be $z$ Since we have three variables, we need to set up a system of three equations. We know the farmer has 320 acres, and he is going to use all of them, so our first equation can be $x+y+z=320$ We also know that the farmer wants to plant twice as many acres of tomatoes as acres of sunflowers, so our second equation can be $2y=z$ We also know that the farmer wants to plant $40$ more acres of corn than tomatoes, so our third equation can be $y=x-40$ Now we have our system of equations: $x+y+z=320$ $2y=z$ $y=x-40$ We can substitute the third equation into the second and first equation to reduce the number of variables from three to two. $x+(x-40)+z=320$ $2(x-40)=2x-80=z$ Since $z$ is already isolated, we can substitue the second equation into the first equation to reduce our system from two variables to one variable. $x+(x-40)+(2x-80)=320$ Now reduce and combine like terms to solve for $x$ $x+(x-40)+(2x-80)=320$ $x+x-40+2x-80=320$ $x+x+2x-40-80=320$ $4x-120=320$ $4x=440$ $x=110$ Now substitute $x$ into our original third equation to solve for $y$ $y=(110)-40$ $y=70$ Now substitute $y$ into our original second equation to solve for $z$ $2(70)=z$ $z=140$ To recap: $x=110$ - the number of acres of corn $y=70$ - the number of acres of tomatoes $z=140$ - the number of acres of sunflowers