C ollege A lgebra

1. University of Palestine IT-College College Algebra Systems and Matrices(Chapter5) L:19 1

2. 5.1 Systems of Linear Equations • After completing this Section, you should be able to: • Know if an ordered pair is a solution to a system of linear equations in two variables or not. • Solve a system of linear equations in two variables by graphing. • Solve a system of linear equations in two variables by the substitution method. • Solve a system of linear equations in two variables by the elimination by addition method. • Solve a system of linear equations in three variables by the elimination method.

3. Linear Systems • Any equation of the form a1x1 + a2x2 +    + anxn = b, for all real numbers a1,a2,…,an (not all of which are 0) and b, is a linear equation or a first-degree equation in n unknowns. • A set of equations is called a systems of equations. The solutions of a system of equations must satisfy every equation in the system. If all the equations in a system are linear, the system is a system of linear equations, or a linear system.

4. Solution of a System • There are three possible outcomes that you may encounter when working with these systems: • one solution • no solution • infinite solutions

5. Consistent The graphs of two equations intersect a single point. The coordinates of this point give the only solution of the system. One Solution A consistent system is a system that has at least one solution.

6. Inconsistent The graphs are distinct parallel lines. The equations are independent. That is, there is no solution common to both equations. No Solution An inconsistent system is a system that has no solution.

7. Dependent The graphs are the same line. Any solution of one equation is also the solution of the other. Thus there are infinite number of solutions. Infinite Solutions The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other equation.  In other words, they end up being the same line.

8. Three Ways to Solve Systems of Linear Equations in Two Variables • There are three ways to solve systems of linear equations in two variables: • graphing  • substitution method • elimination method .

9. Solve by Graphing Example :  Solve the system of equation by graphing. The solution to this system is (2, 1).

10. Substitution Method In a system of two equations with two variables, the substitution method involves using one equation to find an expression for one variable in terms of the other, then substituting into the other equation of the system. Example: Solve the system. 4x + 2y = 8 (1) 3x 7y = 11 (2)

11. Begin by solving one of the equations for one of the variables. 4x + 2y = 8 2y = 4x + 8 y = 2x + 4 (3) Now replace y with 2x + 4 in the second equation and solve for x. 3x  7(2x + 4) = 11 3x + 14x  28 = 11 17x = 17 x = 1 Replace x with 1 in equation (3) to obtain y = 2(1) + 4 = 2. The solution of the ordered pair is (1,2). Check the solution in both equations (1) and (2). 3x 7y = 11 3(1) 7(2) = 11 11 =  11 4x + 2y = 8 4(1) + 2(2) = 8 8 = 8 Solution 4x + 2y = 8 (1) 3x 7y = 11 (2)

12. Solve by the Elimination by Addition Method • The elimination method uses multiplication and addition to eliminate a variable from one equation. To eliminate a variable, the coefficients of that variable in the two equations must be additive inverses. To achieve this, we use properties of algebra to change the system to an equivalent system, one with the same solution set.

13. Equivalent Systems Transformations of a Linear System • Interchange any two equations of the system. • Multiply or divide any equation of the system by a nonzero real number. • Replace any equation of the system by the sum of that equation and a multiple of another equation in the system.

14. Example Solve the system using the elimination method. 6x + 2y = 4 10x + 7y = 8 If we multiply the first equation by 5 and the second equation by 3, we will be able to eliminate the x variable. 30x + 10y = 20 Substituting: 6x + 2y = 4 30x  21y = 24 6x + 2(4) = 4  11y = 44 6x  8 = 4 y =  4 6x = 12 The solution is (2,  4) x = 2

15. Solving an Applied Problem by Writing a System of Equations • Step 1Read the problem carefully until you understand what is given and what is to be found. • Step 2Assign variables to represent the unknown values, using diagrams or tables as needed. Write down what each variable represents. • Step 3Write a system of equations that relates the unknowns. • Step 4Solve the system of equations. • Step 5State the answer to the problem. Does it seem reasonable? • Step 6Check the answer in the words of the original problem.

16. Solving Systems with Three Unknowns (Variables) • The graph of a linear equation in three unknowns requires a three-dimensional coordinate system. Some of the possible intersections of planes representing three equations are shown below.

17. Systems of Three Equations with Three Variables • To solve a system with three unknowns, first eliminate a variable from any two of the equations. Then eliminate the same variable from a different pair of equations. Eliminate a second variable using the resulting two equations in two variables to get an equation with just one variable whose value you can now determine. Find the values of the remaining variables by substitution. Solutions of the systems are written as ordered triples. • Example: Solve the system. 3x + 9y + 6z = 3 (1) 2x + y z = 2 (2) x + y + z = 2 (3)

18. Eliminate z by adding equations (2) and (3) to get 3x + 2y = 4 (4) To eliminate z from another pair of equations, multiply both sides of equations (2) by 6 and add the result to equation (1). 3x + 9y + 6z = 3 12x + 6y 6z = 12 (1) 15x + 15y = 15 (5) To eliminate x from the equations (4) and (5), multiply both sides of equation (4) by 5 and add the result to equation (5). Solve the new equation for y. 15x  10y = 20 15x + 15y = 15 5y = 5 y = 1 Solution 3x + 9y + 6z = 3 (1) 2x + y z = 2 (2)x + y + z = 2 (3)

19. Using y = 1, find x from equation (4) by substitution. 3x + 2(1) = 4 (4) x = 2 Substitute 2 for x and 1 for y in equation (3) to find z. 2 + (1 ) + z = 2 (3) z = 1 Verify that the ordered triple (2, 1, 1) satisfies all three equations in the original system. The solution set is {(2, 1, 1)}. Solution continued

20. Using Curve Fitting to Find an Equation Through Three Points Example: Find the equation of the parabola y = ax2 + bx + c that passes through (2,4), (1, 1), and (2,5). Solution: Since the three points lie on the graph of the equation y = ax2 + bx + c, they must satisfy the equation. Substituting each ordered pair into the equation gives three equations with three variables. 4 = a(2)2 + b (2) + c or 4 = 4a + 2b + c (1) 1 = a(1)2 + b(1) + c or 1 = a  b + c (2) 5 = a(2)2 + b(2) + c or 5 = 4a  2b + c (3)

21. This system can be solved by the elimination method. First eliminate c using equations (1) and (2). 4 = 4a + 2b + c (1) 1 = a + b  c 4 = 3a + 3b (4) Now, use equations (2) and (3) to eliminate the same variable (c). 1 = a  b + c (2) 5 = 4a + 2b  c 4 = 3a + b (5) Solving systems of equations (4) and (5) in two variables by eliminating a. 3 = 3a + 3b (4) 4 = 3a + b (5) 1 = 4b Find a by substituting for b in equation (4), which is equivalent to 1 = a + b. 1 = a + b 1 = a Using Curve Fitting to Find an Equation Through Three Points continued

22. Using Curve Fitting to Find an Equation Through Three Points continued • Finally, find c by substituting a = and b = in equation (2). • An equation of the parabola is

23. 5.2 Matrix Solution of Linear Systems

24. Definitions • A matrix is a rectangular array of numbers enclosed in brackets. • Each number is called an element of the matrix. • The size of a matrix is determined by the number of row (horizontal) and columns (vertical).

25. Linear System Augmented matrix rows columns Definitions continued

26. 5.3 Determinant Solution of Linear Systems • Objectives • Evaluate 2 by 2 and 3 by 3Determinants • Use Cramer’s Rule to Solve a System of Two or Three Equations With Two or Three Variables • Know Properties of Determinants

27. Determinants • Every n n matrix A is associated with a real number called the determinant, of A, written |A|. The determinant of a 2  2 matrix is defined as follows. • Note: Matrices are enclosed with square brackets, while determinants are denoted with vertical bars.

28. Example • Let Find |A|.

29. Determinant of a 3  3 Matrix

30. A 3 by 3 determinant is symbolized by

32. When evaluating a determinant, you can expand across any row or down any column you choose.

33. Cofactor • Let Mij be the minor for the element aij in an n n matrix. The cofactor if aij written Aij, is • Find the cofactor of the element (2). • The cofactor is

34. Finding the Determinant of a Matrix • Multiply each element in any row or column of the matrix by its cofactor. The sum of these products give the value of the determinant. Example • Evaluate expanding by the third row.

35. Example continued • Now find the cofactor of each element of these minors. • Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.

37. Cramer’s Rule for Two Equations in Two Variables

38. Use Cramer’s Rule to solve the system. 7x + 3y = 15 2x + 9y = 12 Find D first, since if D = 0, Cramer’s Rule does not apply. The solution set is {(3, 2)}. Example

39. General Form of Cramer’s Rule

40. Use Cramer’s Rule to solve the system. Example

41. Example continued • Thus: • The solution set is

42. Example

44. Solution: x = 2, y = -1, z = 3

45. 5.4 Properties of Matrices

46. Basic Definitions • It is customary to use capital letters to name matrices. Also, subscript notation is often used to name elements of a matrix, as shown. • A n n matrix is a square matrix. • A matrix with just one row is a row matrix. • A matrix with just one column is a column matrix. • Two matrices are equal if they are the same size and if corresponding elements, position by position, are equal.

47. Example • Find the values of the variables which makes the statement true. • From the definition of equality, the only way that the statement can be true is if a = 3, b = 4, x = 2 and y = 7.

48. Addition and Subtraction of Matrices • To add two matrices of the same size, add corresponding elements. Only matrices of the same size can be added. • If A and B are two matrices of the same size, then A B = A + (B).

49. Examples • Add and subtract the following. • AddSubtract

50. Examples continued • Add or subtract, if possible. • a) • b) The matrices have different sizes so they cannot be added or subtracted.