Solving and Application Using A System with Three Equations

Example

An animal feed is made from three ingredients: corn, soybeans, and cottonseed. One unit of each ingredient provides units of protein, fat, and fiber as shown in the table. How many units of each ingredient should be used to make a feed that contains 22 units of protein, 28 units of fat, and 18 units of fiber?

	Corn	Soybeans	Cottonseed	Total
Protein	0.25	0.40	0.20	22
Fat	0.40	0.20	0.30	28
Fiber	0.30	0.20	0.10	18

Solution

1. Read the problem. We must determine the number of units of corn, soybeans, and cottonseed.

2. Assign variables. Let x represent the number of units of corn, y the number of units of soybeans, and z the number of units of cottonseed.

3. Write a system of equations. The total amount of protein is to be 22 units, so we use the first row of the table to write our first equation.
   $0.25x+0.4y+0.2z=22$

   We use the second row of the table to obtain 28 units of fat.
   $0.4x+0.2y+0.3z=28$

   Finally, we use the third row of the table to obtain 18 units of fiber,
   $0.3x+0.2y+0.1z=18$

   Multiply our first equation on both sides by 100, and the last two equations by 10 to get an equivalent system.
   $25x+40y+20z=2200$
   $4x+2y+3z=280$
   $3x+2y+z=180$

4. Solve the system. Using the methods described earlier in this section, we find that  x = 40, y = 15, and z = 30.

5. State the answer. The feed should contain 40 units of corn, 15 units of soybeans, and 30 units of cottonseed.

6.  Check. Show that the ordered triple (40, 15, 30) satisfies the system formed by the three equations.

Note

Notice how the table in this example is used to set up the equations of the system. The coefficients in each equation are read from left to right. This idea is extended in the next section, where we introduce solution of systems by matrices.

Now please turn to the next page and complete Homework 9.1.