Download presentation

Presentation is loading. Please wait.

Published byDaisy Masser Modified over 2 years ago

1
PRECALCULUS I SOLVING SYSTEMS OF EQUATIONS Dr. Claude S. Moore Cape Fear Community College Chapter 8

2
PRECALCULUS I TWO-VARIABLE LINEAR SYSTEMS 672

3
GRAPHICAL METHOD 1. Graph each equation on the same coordinate (x-y) plane. 2. Find the point(s) of intersection, if any exist. 3. Check the solution(s) in each of the original equations. 672

4
Number of solutions 1. Exactly one consistent; independent 2. Infinitely many consistent; dependent 3. No solution inconsistent Graph interpretations 1. Intersect in one point 2. Lines are identical 3. Lines are parallel INTERPRETING GRAPHS of two linear equations in two variables: 673

5
GRAPHICAL: EXAMPLE 1 Graph and solve (1) 8x + 9y = 42 (2) 6x - y = 16 (1) (0,42/9); (42/8,0) (2) (0,-16); (16/6,0) The solution is (3,2) which checks in both equations. 21 673

6
ELIMINATION METHOD 1. Get coefficients of x (or y) to be opposites of each other. 2. Add equations to eliminate variable. 3. Back-substitute into either equation. 4. Check your solution in both of the original equations. 674

7
ELIMINATION: EXAMPLE 2 Solve by elimination: (1)5u + 6v = 32 (2)3u + 5v = 22 -3(1) -15u - 18v = - 96 5(2) 15u + 25v = 110 (3) 7v = 14 v = 2 In equation (2), substitute v = 2: 3u + 5(2) = 22 3u + 10 = 22 3u = 12 u = 4 Solution is (4,2). 675

8
PRECALCULUS I MULTIVARIABLE LINEAR SYSTEMS 688

9
ROW-ECHELON FORM The Gausian elimination process was named for Carl Friedrich Gauss (1777-1855) a German mathematician who developed the row-echelon form. 688

10
ROW-ECHELON FORM x - 2y + 3z = 9 -x + 3y = - 4 2x - 5y + 5z = 17 This system is in its original form. x - 2y + 3z = 9 y + 3z = 5 z = 2 This equivalent system is in Row-Echelon form. ?

11
ROW OPERATIONS: EQUIVALENT SYSTEMS 1. Interchange two equations. 2. Multiply one equation by non-zero constant. 3. Add a non-zero multiple of one equation to a non-zero multiple of another equation. 688

12
INCONSISTENT SYSTEM Solve the system of equations : E13x - 2y - 6z = -4 E2 -3x + 2y + 6z = 1 E3 x - y - 5z = -3 e1 x - y - 5z = -3 e2 3x - 2y - 6z = -4 e3 -3x + 2y + 6z = 1 e1 x - y - 5z = -3 -3e1+e2 y + 9z = 5 3e1+e3 - 1y - 9z = -8 e1 x - y - 5z = -3 e2 y + 9z = 5 e2+e3 0z = -3 Since 0 = -3 is never true, there is no solution. 691

13
INFINITE SOLUTION Solve the system of equations : E1 x + 2y - 7z = -4 E2 2x + y + z = 13 E3 3x + 9y -36z = -33 Since 0z = 0 is always true, the solution (-3a+10, 5a-7, a) is infinite. e1 x + 2y - 7z = -4 e2-2e1 -3y + 15z = 21 e3-3e1 3y - 15z = -21 e1 x + 2y - 7z = -4 (-1/3)e2 y - 5z = -7 e3+e1 0z = 0 691

14
NON-SQUARE SYSTEM Solve the system of equations : E1 x - 3y + 2z = 18 E2 5x - 13y + 12z = 80 Substitute z = a and y = -a -5 into equation 1 and get the solution (-5a + 3, -a - 5, a). e1 x - 3y + 2z = 18 -5e1+e2 2y + 2z = -10 e1 x - 3y + 2z = 18 (1/2)e2 y + z = -5 So y = -z - 5. Let z =a. Then y = -a - 5. ?

15
PRECALCULUS I TWO-VARIABLE NON-LINEAR SYSTEMS 672

16
SUBSTITUTION METHOD 1. Solve one equation for one variable. 2. Substitute into other equation. 3. Solve equation from Step 2. 4. Back-substitute into Step 1. 5. Check the solution in each equation. Section 9-4

17
SUBSTITUTION: EXAMPLE 1 Solve x + y = 0 and x 3 - 5x - y = 0. Substitute y = -x to get x 3 - 5x - (-x) = 0 x 3 - 5x + x = 0 or x 3 - 4x = 0 x(x 2 - 4) = 0 or x(x + 2)(x - 2) = 0 Thus x = 0, x = -2, and x = 2 giving the solutions (-2,2), (0,0), and (2,-2). Section 9-4

18
GRAPHICAL METHOD 1. Graph each equation on the same coordinate (x-y) plane. 2. Find the point(s) of intersection, if any exist. 3. Check the solution(s) in each of the original equations. Section 9-4

19
GRAPHICAL: EXAMPLE 2 Graph and solve y = e x & x - y = -1 x - y = -1 yields y = x + 1. Solution is (0,1) which checks in both equations: 1 = e 0 and 0 - 1 = -1. Section 9-4

20
GRAPHICAL: EXAMPLE 3 Graph and solve (1) 2x - y = 1 (2) x 2 + y = 2 (1) y = 2x - 1 (2) y = -x 2 + 2. Solutions are (-3,-7) & (1,1) which check in both equations. Section 9-4

Similar presentations

OK

Chapter 4 Systems of Linear Equations; Matrices Section 2 Systems of Linear Equations and Augmented Matrics.

Chapter 4 Systems of Linear Equations; Matrices Section 2 Systems of Linear Equations and Augmented Matrics.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on motivation concepts Ppt on bluetooth architecture overview Ppt on culture and science in the ancient period of philosophy Signal generator and display ppt online Ppt on second law of thermodynamics definition Ppt on any one mathematicians Ppt on effect of global warming on weather service Ppt on therapeutic environment in nursing Ppt on bridge construction in india Ppt on emotional intelligence in leadership