1. Copyright © 2011 Pearson Education, Inc. Slide 7.2-1

2. Copyright © 2011 Pearson Education, Inc. Slide 7.2-2 Chapter 7: Systems of Equations and Inequalities; Matrices 7.1Systems of Equations 7.2Solution of Linear Systems in Three Variables 7.3Solution of Linear Systems by Row Transformations 7.4Matrix Properties and Operations 7.5Determinants and Cramer’s Rule 7.6Solution of Linear Systems by Matrix Inverses 7.7Systems of Inequalities and Linear Programming 7.8Partial Fractions

3. Copyright © 2011 Pearson Education, Inc. Slide 7.2-3 7.2 Solution of Linear Systems in Three Variables Solutions of systems with 3 variables with linear equations of the form Ax + By + Cz = D(a plane in 3-D space) are called ordered triples (x, y, z). Possible solutions:

4. Copyright © 2011 Pearson Education, Inc. Slide 7.2-4 7.2 Solving a System of Three Equations in Three Variables Solve the system. Eliminate z by adding (2) and (3) 3x + 2y = 4(4) Eliminate z from another pair of equations, multiply (2) by 6 and add the result to (1). Eliminate x from equations 4 and 5. Multiply (4) by -5 and add to (5).

5. Copyright © 2011 Pearson Education, Inc. Slide 7.2-5 7.2 Solving a System of Three Equations in Three Variables continued Using y = –1, find x from equation (4) by substitution. 3x + 2(–1) = 4 x = 2 Substitute 2 for x and –1 for y in equation (3) to find z. 2 + (–1) + z = 2 z = 1 The solution set is {(2, –1, 1)}.

6. Copyright © 2011 Pearson Education, Inc. Slide 7.2-6 7.2 Solving a System of Two Equations and Three Unknowns ExampleSolve the system. SolutionGeometrically, the solution of two non- parallel planes is a line. Thus, there will be an infinite number of ordered triples in the solution set.

7. Copyright © 2011 Pearson Education, Inc. Slide 7.2-7 7.2 Solving a System of Two Equations and Three Unknowns This is as far as we can go with the echelon method. Solve y + z = 3 to get y = 3 – z for any arbitrary value for z. Now we express x in terms of z by solving equation (1).

8. Copyright © 2011 Pearson Education, Inc. Slide 7.2-8 7.2 Solving a System of Two Equations and Three Unknowns The solution set is written {(z – 2, 3 – z, z)}. For example, if z = 1, then y = 3 – 1 = 2 and x = 1 – 2 = –1, giving the solution set {(–1, 2, 1)}. Verify that another solution is {(0, 1, 2)}. Let y = 3 – z.

9. Copyright © 2011 Pearson Education, Inc. Slide 7.2-9 7.2 Application: Solving a System of Three Equations to Satisfy Feed Requirements 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?

10. Copyright © 2011 Pearson Education, Inc. Slide 7.2-10 7.2 Application: Solving a System of Three Equations to Satisfy Feed Requirements SolutionLet x represent the number of units of corn, y, the number of units of soybeans, and z, the number of units of cottonseed. Using the table, we get the following system or, We can show that x = 40, y = 15, and z = 30.

11. Copyright © 2011 Pearson Education, Inc. Slide 7.2-11 7.2 Curve Fitting Using A System Find the equation of the parabola with vertical axis that passes through the points (2, 4), (  1, 1), and (  2, 5). Substitute each ordered pair into the equation ax 2 + bx + c. 4 = 4a + 2b + c(1) 1 = a – b + c(2) 5 = 4a – 2b + c (3) Eliminate c using equations (1) and (2). Eliminate c using equations (2) and (3).

12. Copyright © 2011 Pearson Education, Inc. Slide 7.2-12 7.2 Curve Fitting Using A System continued Solve the system of equations in two variables by eliminating a. Find a by substituting for Find c by substituting b in equation (4). for a and b in equation (2) Find c by substituting a and b in equation (2). The equation of the parabola is: