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Pre Calculus Chapter 8

Pre Calculus Chapter 8. Systems of Equations and Inequalities. Systems of Equations—8.1. Substitution Method. Solve for 1 variable Substitute this value or equation into the other equation Find an answer Plug this answer into the other equation to find your solution. Substitution.

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Pre Calculus Chapter 8

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  1. Pre Calculus Chapter 8 Systems of Equations and Inequalities

  2. Systems of Equations—8.1

  3. Substitution Method • Solve for 1 variable • Substitute this value or equation into the other equation • Find an answer • Plug this answer into the other equation to find your solution

  4. Substitution

  5. Substitution

  6. Elimination (Combination) Method • Multiply 1 or more equations by some constant to get coefficients to cancel • Add/subtract equations as necessary • Back Substitute • Smile 

  7. Elimination

  8. Elimination

  9. p.628#1-4, 9-11, 17-20, 41-42, 46

  10. Systems of Linear Equations in Two Variables—8.2

  11. Number of Solutions • The system can have exactly 1 solution • The system can have no solution • The system can have infinitely many solutions

  12. Number of Solutions • A system that has no solution is said to be inconsistent • A system with infinitely many solutions is called dependent

  13. Algebraically

  14. Algebraically

  15. Modeling With Equations • A woman rows a boat upstream from one point to another point 4 miles away in 1 ½ hours. The return trip, downstream, takes 45 minutes. How fast is she rowing relative to the water and at what speed is the current flowing?

  16. Modeling With Equations • A woman rows a boat upstream from one point to another point 4 miles away in 1 ½ hours. The return trip, downstream, takes 45 minutes. How fast is she rowing relative to the water and at what speed is the current flowing?

  17. Modeling • A woman rows a boat upstream from one point to another point 4 miles away in 1 ½ hours. The return trip, downstream, takes 45 minutes. How fast is she rowing relative to the water and at what speed is the current flowing?

  18. Modeling • A biologist has two brine solutions, one containing 5% salt and another containing 20% salt. How many milliliters of each solution should he mix to obtain 1 L of a solution that is 14% salt?

  19. p.635# 7-9, 20-22, 35, 37-40

  20. Systems of Linear Equations in Several Variables—8.3

  21. Solving • Same methods as any system of equations • Goal: Triangular form

  22. What Can We do? • Add a nonzero multiple of one equation to another. • Multiply an equation by a nonzero constant • Interchange the positions of two equations

  23. Solving

  24. Systems With Alternate Solutions

  25. Systems With Alternate Solutions

  26. Systems With Alternate Solutions

  27. Applications • You receive an inheritance of $50,000. You would like to invest all of this money into three different funds: Money-market fund, blue-chip stock, and high-tech stock. Your broker estimates the MM will earn 5% this year, the BC 9%, and the HT 16%. Your goal is to get a $4000 return the first year. However, to avoid excess risk, you decide to invest 3 times more in the MM as the HT. How much do you need to invest in each fund?

  28. Curve Fitting • We know that any quadratic can be written in the form: • Since we are finding 1 equation for 3 points, we use the same a, b, and c values

  29. Curve Fitting • Find the equation of the quadratic that passes through the following points:

  30. p.646#5-6, 15-17, 33-35, 37-8 (10 point)

  31. Systems of Linear Equations and Matrices—8.4

  32. What is a Matrix? • Matrix • Rectangular array of numbers • Dimensions • Rows x Columns • Entries • Elements

  33. Equality of Matrices • Two matrices are equal iff: • They have the same dimensions • Corresponding entries are equal

  34. Matrices • An m x n matrix is an array of numbers with m rows and n columns

  35. Matrices

  36. Matrices • Row Operations: • Add a multiple of one row to another • Multiply a row by a non-zero constant • Interchange 2 rows

  37. Matrices • Solve the following linear system:

  38. Row-echelon form (REF) • First non-zero number in each row is 1 • Triangle Form • Any all zero rows are at bottom of matrix

  39. Reduced row-echelon form (RREF) • Every Number above and below each leading entry is a 0

  40. Solutions to Matrices • No Solution • One Solution • Infinite Solutions

  41. p.659#1-3, 9-11, 19-22, 35 (10 pt)

  42. Algebra of Matrices—8.5

  43. Algebra of Matrices • Addition: • Equal dimensions • Add corresponding entries • Difference of matrices • Subtract corresponding entries • Scalar Product • Multiplying a matrix by a constant • Multiply each element by the scalar

  44. Addition

  45. Adding and Subtracting Matrices • Example:

  46. Subtraction

  47. Adding and Subtracting Matrices • Example:

  48. Scalar Multiplication • Any matrix can be multiplied by a real number constant, called a scalar • This is called scalar multiplication

  49. Scalar Multiplication • Example:

  50. Scalar Multiplication

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