1. WEEK 6 Day 2

2. Progress report Thursday the 11 th.

4. Objectives Solve systems of equations by substitution. Solve systems of equations by the addition- subtraction method. Evaluate determinants using determinant properties. Use Cramer’s rule. Use the method of partial fractions to rewrite rational expressions as the sum or the difference of simpler expressions.

5. 6.1 SOLVING A SYSTEM OF TWO LINEAR EQUATIONS page212 In this section we shall study solutions by: 1. Graphing 2. Addition-subtraction method 3. Method of substitution

6. 6.1 SOLVING A SYSTEM OF TWO LINEAR EQUATIONS page213 Any ordered pair (x, y) that satisfies both equations is called a solution, or root, of the system.

7. Method 1 Graphing (plot) page 213 When the two lines intersect, the system of equations is called independent and consistent.

8. Page 213 When the two lines are parallel, the system of equations is called inconsistent.

9. Page 213 When the two lines coincide, the system of equations is called dependent.

10. Method 1 Graphing (plot) page 213

11. Method 2 Add and Subtract page 214 The first algebraic method (second method over all) is called the: addition-subtraction method. (sometimes called the elimination method) eliminating X or Y

12. Page 214 multiply each side of one or both equations by some number so that the numerical coefficients of one of the variables are of equal absolute value.

13. Addition Subtraction 2x + 3y = -4 X – 2y = 5 2 (X – 2y = 5) 2 2x – 4y = 10 Only 1 equation but both sides.

14. 2x + 3y = -4 -2x – 4y = 10 7y = - 14 7 7 Y = -2 No “x”.

15. Substitute y = -2 in either original equation. 2 x + 3(-2) = -4 2x + -6 = -4 2x = -4 + 6 2 x = 2 x = 1

16. multiply each side of one or both equations by some number 2x + 3y = -4 X – 2y = 5 2 2x + 3y = -4 2 4x + 6y = -8 4 (X – 2y = 5) 4 4x – 8y = 20 Both equations.

17. multiply each side of one or both equations by some number 4x + 6y = -8 -(4x – 8y) = 20 14y = - 28 14 14 y = -2 No “x”.

18. Check by substituting

19. Method 3 Substitution Page 215 The second algebraic method (3 rd method over all) of solving systems of linear equations is called the method of substitution.

20. Page 215 3x + y = 3 2x - 4y = 16 Solve for x or y. (y)3x + y = 3 y = -3x + 3

21. Page 215 2x - 4y = 162x – 4(-3x + 3) = 16 2x + 12x - 12 = 16 14x = 28 x = 2 No “y”.

22. Page 217 A special case of the substitution method is the comparison method: a = c b = c a = b

23. Page 217 Comparison method: 3x – 4 = 5y 6 – 2x = 5y Since the left side of each equation equals the same quantity, we have: 3x – 4 = 6 – 2x This eliminates the variable y.

24. Page 218 Section 6.1

26. 6.2 OTHER SYSTEMS OF EQUATIONS A literal equation is one in which letter coefficients are used in place of numerical coefficients. No numbers.

27. In the equations, a and b represent known quantities or coefficients, and x and y are the variables or unknown quantities.

28. 6.2 OTHER SYSTEMS OF EQUATIONS page 222

34. Substitute b for x.ax + by = ab a(b) + by = ab - a(b) = - ab by = 0 x = b and y = 0 (b, y)

35. End Week 6 Day 1

36. 6.2 The equations in the system are not linear, or first-degree, equations.

37. The Lowest Common Denominator is: xy

38. 6y + 4x = -2xy

40. 6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS Page 224

41. 6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 224 The graph of a linear equation with three variables in the form is a plane.

42. Graphical solutions of three linear equations with three unknowns are not used because three-dimensional graphing is required and is not practical by hand.

43. 6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 225 Let’s choose to eliminate x first. To eliminate x from any pair of equations, such as (1) and (2), multiply each side of Equation by your chosen number and subtract.

44. 6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 225 Choosing to multiply each side of Equation (1) by 2 you get. (2)

45. 6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 225 To eliminate x from any other pair of equations, such as (1) and (3), multiply each side of Equation (1) by 3 and add. (3)

46. 6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 226 We have now reduced the system of three equations in three variables to a system of two equations in two variables,

47. 6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 226 Substitute 2 for z in one of the equations and solve for y.

48. Review

49. A product is the result obtained by multiplying two or more quantities together. Factoring is finding the numbers or expressions that multiply together to make a given number or equation.

50. ProductFactoring

51. 5.4 EQUIVALENT FRACTIONS page 189 Two fractions are equivalent when both the numerator and the denominator of one fraction can be multiplied or divided by the same nonzero number.

52. 5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195

54. Reorganize like terms.

55. Factor each of the terms in the numerator and denominator. Divide by common factors. 60 a y 30 b x

56. 5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195

57. Page 196

60. In class exercise week 6 day 2.