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Download 3.5 Solving Systems of Equations in Three Variables
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7.3 Solving Systems of Equations in Three Variables Or when planes crash together So far we have solved for the intersection of lines Do you remember what you get when planes intersect? So far we have solve for the intersection of lines Did you remember what you get when planes intersect? You form lines What happens when you intersect 3 planes? What happens when you intersect 3 planes? You sometimes get points with three variables. What happens when you intersect 3 planes? You sometimes get points with three variables. Of course they can intersect in different ways. Here we get a line again. What happens when you intersect 3 planes? You sometimes get points with three variables. Of course they can intersect in different ways. Of course we can get nothing. This would be No solution. You could just have three planes that do not intersect at all Parallel planes. Solve the system of equations by Gaussian Elimination What is Gaussian Elimination? In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations.Gauss – Jordan elimination, an extension of this algorithm, reduces the matrix further to diagonal form, which is also known as reduced row echelon form. http://en.wikipedia.org/wiki/Gaussian_elimination Solve the system of equations by Gaussian Elimination 5 x 3 y 2 z 2 2x y z 5 x 4 y 2 z 16 I am going to rewrite the system Solve the system of equations by Gaussian Elimination x 4 y 2 z 16 2x y z 5 5 x 3 y 2 z 2 Going to multiply row 1 by -2 and add to row 2 Going to multiply row 1 by -5 and add to row 3 Solve the system of equations by Gaussian Elimination x 4 y 2 z 16 2x y z 5 5 x 3 y 2 z 2 Going to multiply row 1 by -2 and add to row 2 Going to multiply row 1 by -5 and add to row 3 Solve the system of equations by Gaussian Elimination x 4 y 2 z 16 7 y 5z 27 17 y 8z 78 Going to multiply row 2 by (17/-7) and add to row 3 Solve the system of equations by Gaussian Elimination x 4 y 2 z 16 7 y 5z 27 (29 / 7) z (87 / 7) Going to multiply row 2 by (17/-7) and add to row 3 Solve the system of equations by Gaussian Elimination x 4 y 2 z 16 7 y 5z 27 (29 / 7) z (87 / 7) Going to multiply row 3 by (7/29) Solve the system of equations by Gaussian Elimination x 4 y 2 z 16 7 y 5 z 27 z 3 Going to multiply row 3 by -5 and add to row 2 Solve the system of equations by Gaussian Elimination x 4 y 2 z 16 7 y 42 z 3 Going to multiply row 2 by (-1/7) Solve the system of equations by Gaussian Elimination x 4 y 22 y6 z 3 Going to multiply row 3 by -2 and add to row 1 Solve the system of equations by Gaussian Elimination x 4 y 22 y6 z 3 Going to multiply row 2 by -4 and add to row 1 Solve the system of equations by Gaussian Elimination x 2 y6 z 3 Going to multiply row 2 by -4 and add to row 1 Solve the system 5x + 3y + 2z = 2 2x + y – z = 5 x + 4y + 2z = 16 The point of intersect for the system is ( - 2, 6, - 3) These points make all the equations true. Now one with infinite solutions 2x + y – 3z = 5 x + 2y – 4z = 7 6x + 3y – 9z = 15 Middle equation by – 6 added to the third equation. 6x + 3y – 9z = 15 -6x - 12y + 24z = - 42 When added together -9y + 15y = - 27 Solve the new system - 3y + 5z = - 9 -9y + 15z = - 27 Multiply the top equation by – 3 then add to the bottom equation 9y – 15z = 27 -9y + 15z = - 27 0=0 Infinite many solutions One the has no solutions 3x – y – 2z = 4 6x + 4y + 8z = 11 9x + 6y + 12z = - 3 Multiply the first equation by – 2 and add to the middle equation. -6x + 2y + 4z = - 8 6x + 4y + 8z = 11 6y + 12z = 3 One the has no solutions 3x – y – 2z = 4 6x + 4y + 8z = 11 9x + 6y + 12z = - 3 Multiply the first equation by – 3 and add to the last equation. -9x + 3y + 6z = - 12 9x + 6y + 12z = - 3 9y + 18z = - 15 Solve the new system 6y + 12z = 3 multiply by 3 18y + 36z = 9 9y + 18z = - 15 multiply by – 2 -18y – 36z = 30 Add together 18y + 36z = 9 -18y – 36z = 30 0 = 39 Wrong!, No solution. Homework Page 507# 4, 16, 28, 38, 46, 54, 66 Homework Page 507 # 10, 22, 32, 42, 50, 60
