Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

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1 Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an ordered pair (x, y) of real numbers that satisfies each equation in the system. The solutions of the system correspond to the points of intersection of the graphs of the equations. A system of linear equations can have exactly one solution, infinitely many solutions, or no solution. Ex.1 Checking solutions of a system of equations. Check whether each ordered pair is a solution of the system of equations { x + y = 6 2x 5y = 2 (1) (3, 3) (2) (4, 2) 1

2 Ex.2 Use the graphical method to solve the system of equations { 2x + 3y = 7 2x 5y = 1 The method of substitution In order to solve a system of two equations involving two variables you need to follow these steps: (1) Solve one of the equations for one variable in terms of the other. (2) Substitute the expression obtained in Step 1 in the other equation to obtain an equation in one variable. (3) Solve the equation obtained in Step 2. (4) Back-substitute the solution from Step 3 in the expression obtained in Step 1 to find the value of the other variable. (5) Check the solution to see that it satisfies both of the original equations. 2

3 Ex.3 Solve the following system of equations { x + y = 3 3x + y = 1 Ex.4 Solve the following system of equations { 2x 2y = 0 x y = 1 3

4 Ex.5 Find two positive integers such that the sum of the numbers is 8 and their difference is 2. 4

5 Section 4.2: Linear Systems in Two Variables The method of elimination The steps for solving a system of linear equations by the method of elimination are the following: (1) Obtain coefficients for x (or y) that are opposites by multiplying all terms of one or both equations by suitable constants. (2) Add the equations to eliminate one variable, and solve the resulting equation. (3) Back-substitute the value obtained in Step 2 in either the original equations and solve for the other variable. (4) Check your solution in both of the original equations. Ex.1 Solve the following system of linear equations { 3x + 2y = 4 5x 2y = 8 5

6 Ex.2 Solve the following system of linear equations { 4x 5y = 13 3x y = 7 Ex.3 Solve the following system of linear equations { 3x + 9y = 8 2x + 6y = 7 6

7 Ex.4 Solve the following system of linear equations { 2x + 6y = 3 4x 12y = 6 7

8 Section 4.3: Linear Systems in Three Variables Ex.1 Solve the following system of linear equations x 2y + 2z = 9 y + 2z = 5 z = 3 The method of Gauss elimination Two systems of equations are equivalent if they have the same solution set. To solve a system that is not in row-echelon form, first convert it to an equivalent system that is in row-echelon form and then solve the new system. Each of the following row operations on a system of linear equations produces an equivalent system of linear equations. (1) Interchange two equations. (2) Multiply one of the equations by a nonzero constant. (3) Add a multiple of one of the equations to another equation to replace the latter equation. Ex.2 Solve the following system of linear equations { 3x 2y = 1 x y = 0 8

9 Ex.3 Solve the following system of linear equations x 2y + 2z = 9 x + 3y = 4 2x 5y + z = 10 9

10 Ex.4 Solve the following system of linear equations 4x + y 3z = 11 2x 3y + 2z = 9 x + y + z = 3 10

11 Ex.5 Solve the following system of linear equations x 3y + z = 1 2x y 2z = 2 x + 2y 3z = 1 11

12 Ex.6 Solve the following system of linear equations x + y 3z = 1 y z = 0 x + 2y = 1 Solutions of a linear system For a system of linear equations, exactly one of the following is true. (1) There is exactly one solution. (2) There are infinitely many solutions. (3) There is no solution. 12

13 Section 4.4: Matrices and Linear Systems Matrices A matrix (plural matrices) is a rectangular array of real numbers, such as An item in a matrix is called an entry or an element. If the matrix has n rows and m columns, its order is n m. A matrix with the same number of rows and columns is called a square matrix. Ex.1 Determine the order of each matrix. (1) ( ) (2) ( 0 ) (3) Augmented matrices A matrix derived from a system of linear equations is the augmented matrix. The matrix derived from the coefficients of the system (but not including the constant terms) is the coefficient matrix of the system. 13

14 Ex.2 Form the coefficient matrix and the augmented matrix for each system. (1) { x + 5y = 2 7x 2y = 6 (2) 3x + 2y z = 1 x + 2z = 3 2x y = 4 Ex.3 Write the system of linear equations represented by each matrix. (1) (2) (3)

15 Elementary row operations As in the case of systems of linear equations there are three elementary row operations that produce equivalent matrices. The elementary row operations are (1) Interchange two rows. (2) Multiply a row by a nonzero constant. (3) Add a multiple of a row to another row. Ex.4 Given the matrix ( ) do the following elementary row operations (1) Interchange the first and second row. (2) Multiply the first row by 3. (3) Add 2 times the first row to the second row. 15

16 Definition of row-echelon form of a matrix A matrix is in row-echelon form if (1) All entries at the bottom of the diagonal are equal to zero. (2) The first nonzero entry in a row is 1 (leading 1). (3) For two consecutive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. Ex.5 The matrix is in row-echelon form Gaussian elimination We can perform the Gaussian elimination using matrices and the following steps: (1) Write the augmented matrix of the system of linear equations. (2) Use elementary row operations to rewrite the augmented matrix in row-echelon form. (3) Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution. 16

17 Ex.6 Solve the following system of linear equations { 2x 3y = 2 x + 2y = 13 17

18 Ex.7 Solve the following system of linear equations 3x + 3y = 9 2x 3z = 10 6y + 4z = 12 18

19 Ex.8 Solve the following system of linear equations { 6x 10y = 4 9x 15y = 5 19

20 Ex.9 Solve the following system of linear equations { 12x 6y = 3 8x + 4y = 2 20

21 Section 4.5: Determinants and Linear Systems Determinant of a 2 2 matrix Given a matrix 2 2 the determinant is ( ) a1 b A = 1 a 2 b 2 det(a) = a 1 b 1 a 2 b 2 = a 1b 2 a 2 b 1 Ex.1 Find the determinant of each matrix. (1) ( 2 3 ) 1 4 (2) ( 1 2 ) 2 4 (3) ( ) 21

22 Determinant of a 3 3 matrix Given a matrix 3 3 the determinant is A = det(a) = a 1 b 2 c 2 b 3 c 3 a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 b 1 a 2 c 2 a 3 c 3 + c 1 a 2 b 2 a 3 b 3 Ex.2 Find the determinant of each matrix. (1) (2)

23 Cramer s rule For the system of linear equations the solution is given by where D 0. For the system of linear equations the solution is given by where D 0. { a1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 x = D x D = y = D y D = c 1 b 1 c 2 b 2 a 1 b 1 a 2 b 2 a 1 c 1 a 2 c 2 a 1 b 1 a 2 b 2 a 1 x + b 1 y + c 1 z = d 1 a 2 x + b 2 y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3 x = D x D = y = D y D = z = D z D = d 1 b 1 c 1 d 2 b 2 c 2 d 3 b 3 c 3 a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 a 1 d 1 c 1 a 2 d 2 c 2 a 3 d 3 c 3 a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 a 1 b 1 d 1 a 2 b 2 d 2 a 3 b 3 d 3 a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 23

24 Ex.3 Use Cramer s rule to solve the following system of linear equations { 4x 2y = 10 3x 5y = 11 24

25 Ex.4 Use Cramer s rule to solve the following system of linear equations x + 2y 3z = 1 2x + z = 0 3x 4y + 4z = 2 25

26 Section 4.6: Systems of Linear Inequalities Graphing a system of linear inequalities (1) Sketch the line that corresponds to each inequality. (2) Lightly shade the half-plane that is the graph of each linear inequality. (3) The graph of the system is the intersection of the half-planes. Ex.1 Sketch the graph of the following system of linear inequalities { 2x y 5 x + 2y 2 26

27 Ex.2 Sketch the graph of the following system of linear inequalities { y < 4 y > 1 27

28 Ex.3 Sketch the graph of the following system of linear inequalities x y < 2 x > 2 y 3 Ex.4 Sketch the graph of the following system of linear inequalities x + y 5 3x + 2y 12 x 0 y 0 28

29 Ex.5 Find a system of inequalities that defines the region shown below. 29

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