883 9.1 Systems of Linear Equations EXAMPLE 9 Solving an Application Using a System of Three Equations 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 0.4 0.2 22 Fat 0.4 0.2 0.3 28 Fiber 0.3 0.2 0.1 18 SOLUTION Step 1 Read the problem. We must determine the number of units of corn, soybeans, and cottonseed. Step 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. Step 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 equation (1). 0.25x + 0.4y + 0.2z = 22 (1) We use the second row of the table to obtain 28 units of fat. 0.4x + 0.2y + 0.3z = 28 (2) Finally, we use the third row of the table to obtain 18 units of fiber. 0.3x + 0.2y + 0.1z = 18 (3) Multiply equation (1) on each side by 100, and equations (2) and (3) by 10, to obtain an equivalent system. 25x + 40y + 20z = 2200 (4) 4x + 2y + 3z = 280 (5) 3x + 2y + z = 180 (6) Step 4 Solve the system. Using the methods described earlier in this section, we find the following. x = 40, y = 15, and z = 30 Step 5 State the answer. The feed should contain 40 units of corn, 15 units of soybeans, and 30 units of cottonseed. Step 6 Check. Show that the ordered triple 140, 15, 302 satisfies the system formed by equations 112, 122, and 132. S Now Try Exercises 109 and 111. Eliminate the decimal points in equations (1), (2), and (3) by multiplying each equation by an appropriate power of 10. NOTE Notice how the table in Example 9 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 the solution of systems by matrices.
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