* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Sec 7.2
Survey
Document related concepts
Transcript
Copyright © 2011 Pearson Education, Inc. Slide 7.2-1 Chapter 7: Systems of Equations and Inequalities; Matrices 7.1 Systems of Equations 7.2 Solution of Linear Systems in Three Variables 7.3 Solution of Linear Systems by Row Transformations 7.4 Matrix Properties and Operations 7.5 Determinants and Cramer’s Rule 7.6 Solution of Linear Systems by Matrix Inverses 7.7 Systems of Inequalities and Linear Programming 7.8 Partial Fractions Copyright © 2011 Pearson Education, Inc. Slide 7.2-2 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: Copyright © 2011 Pearson Education, Inc. Slide 7.2-3 7.2 Solving a System of Three Equations in Three Variables Solve the system. 3x 9 y 6 z 3 2x y z 2 (1) (2) x yz 2 (3) 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). 12 x 6 y 6 z 12 3x 9 y 6 z 3 15 x 15 y 15 (5) Eliminate x from equations 4 and 5. Multiply (4) by -5 and add to (5). 15 x 10 y 10 15 x 15 y 15 5 y 5 y 1 Copyright © 2011 Pearson Education, Inc. Slide 7.2-4 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)}. Copyright © 2011 Pearson Education, Inc. Slide 7.2-5 7.2 Solving a System of Two Equations and Three Unknowns Example Solve the system. x 2y z 4 3x y 4z 9 (1) (2) Solution Geometrically, the solution of two nonparallel planes is a line. Thus, there will be an infinite number of ordered triples in the solution set. 3R1 R 2 R 2 Copyright © 2011 Pearson Education, Inc. x 2y z 4 7 y 7 z 21 (1) Slide 7.2-6 7.2 Solving a System of Two Equations and Three Unknowns 3R1 R 2 R 2 1 R2 R2 7 x 2y z 4 7 y 7 z 21 (1) x 2y z 4 y z 3 (1) 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). Copyright © 2011 Pearson Education, Inc. Slide 7.2-7 7.2 Solving a System of Two Equations and Three Unknowns x 2y z 4 x 2 y z 4 x 2(3 z ) z 4 x z2 Let y = 3 – z. 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)}. Copyright © 2011 Pearson Education, Inc. Slide 7.2-8 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? Copyright © 2011 Pearson Education, Inc. Slide 7.2-9 7.2 Application: Solving a System of Three Equations to Satisfy Feed Requirements Solution Let 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 .25 x .4 y .2 z 22 Protein .4 x .2 y .3z 28 Fat .3x .2 y .1z 18 Fiber, or, 25 x 40 y 20 z 2200 4 x 2 y 3z 280 3x 2 y z 180. We can show that x = 40, y = 15, and z = 30. Copyright © 2011 Pearson Education, Inc. Slide 7.2-10 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 ax2 + bx + c. 4 = 4a + 2b + c (1) 1=a–b+c (2) 5 = 4a – 2b + c (3) 4 4a 2b c Eliminate c using equations (1) and (2). 1 a b c 3 3a 3b Eliminate c using equations (2) and (3). 1 a bc 5 4a 2b c 4 3a b Copyright © 2011 Pearson Education, Inc. Slide 7.2-11 7.2 Curve Fitting Using A System 3 3a 3b continued 4 3a b Solve the system of equations in two 1 4b variables by eliminating a. 1 b 4 Find a by substituting for b in equation (4). 1 a b 1 1 a 4 5 a 4 Find c by substituting for a and b in equation (2) 1 a bc Find c by substituting a and b in equation (2). The equation of the parabola is: y Copyright © 2011 Pearson Education, Inc. 5 1 c 4 4 6 1 c 4 1 c 2 1 5 2 1 1 x x . 4 4 2 Slide 7.2-12