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College Algebra

College Algebra. Acosta/ Karwowski. Chapter 2. Systems of equations. A graphical explanation of systems. Chapter 2 - section 1. Definitions. a system is a set of 2 or more equations that are statements about the same domain and range.

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College Algebra

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  1. College Algebra Acosta/Karwowski

  2. Chapter 2 Systems of equations

  3. A graphical explanation of systems Chapter 2 - section 1

  4. Definitions • a system is a set of 2 or more equations that are statements about the same domain and range. • the solution of a system is the point or points at which both statements are true - this is the mathematical definition of intersection. • Therefore the solution of a system is where the graphs of the 2 equations intersect. • If the equations are linear this can happen once, never, or everywhere

  5. What the graphs tell us • consistent and inconsistent • independent and dependent g(x) j(x) k(x) f(x) m(x) n(x)

  6. Using graphs to solve a system • Given a graph the solution is the point of intersection

  7. Using systems to solve linear equations in one variable • An equation in one variable can be viewed as a system • Ex: 3(x – 5) + 2x = 7x – 9 becomes f(x) = 3(x – 5) + 2x and g(x) = 7x – 9 When the solution on the screen appears the value for x is the solution to the equation

  8. Solving numerically • Before technology this was called guess and check • now it is called using a table to solve • given 2 equations you generate a table with 3 columns when you graph. The Solution is the place where column 2 is the same as column 3

  9. Examples • Find the solution for the following system

  10. Using your graphing calculator • Solve equations for y • Enter the equations in calculator • Press graph - adjust window so the intersection shows (for linear equations this isn’t really necessary) • Press 2nd button and trace button – choose number 5 in the menu. • a cursor appears on one of the lines – use arrows to move the cursor close to the point of intersection (with linear equations this is not necessary but it IS necessary when we do non- linear systems) • the calculator is displaying the equation for the line at the top and a question at the bottom. Press enter to answer yes - another question appears – press enter again- ANOTHER question appears – press enter AGAIN. (in other words press enter 3 times) • now there are 2 numbers displayed at the bottom of the screen. This is the point of intersection

  11. Using technology • Solve both equations for y • Enter the equations • Press 2nd button and graph button and a table appears. Use the up and down arrows to scroll through the table or go to table settings and choose a different start number • if you can’t find a match you have to go to table set and choose a new • This is EXTREMELY tedious -

  12. Major draw back for this method • the calculator will give you decimal values and if they are repeating decimals or irrational numbers this may not be accurate enough.

  13. Solving 2 variable systems Chapter 2 – section 2

  14. Underlying theory • If there are 2 variables in the problem I have infinite solutions • If I can eliminate one of the variables I get an equation with one solution • The second equation gives me a way to eliminate one of the variables • There are 2 pieces of mathematics that accomplish this One is called substitution The other is called elimination or sometimes addition and multiplication method

  15. Substitution • Substitution is replacing one “number” with another form of the number that is equal to it • You do this when you evaluate a function Ex: find 3x = y when x = -5 • You can do this with full equations also

  16. Examples • y = 3x + 2 3y – 2x = 8 • 3x – y = 26 3y + 2x = -1 • 2x + 3y = 12 6y = 15 – 4x • f(x) = 3x + 8 g(x) = 7 – 2x • 2x – 3y = 12 3x + 6y = 32

  17. Elimination • Sometimes simply called the addition/multiplication method because that is what you do • For any equation you can add anything you want as long as you add to both sides • This includes adding equations - as long as you add both sides • This is helpful if one of the variable terms match - If they don’t you have to multiply to make them match

  18. examples • 2x + 4y = 26 -2x – 6y = -32 • x – y = 14 x + 4y = 10 • 2x + 3y = 19 5x – 4y = -10 • 3x = -5 – 7y 2y + 6x = 14 • 2x + 4y = 3 8y – x = 11 • 2(x – 8) + 3y = -18 y – 3x = 25

  19. More examples

  20. 3 variable systems Chapter 2-section 2-b

  21. Systems in 3 variables • The system requires 3 equations in order to solve. • Geometrically each equation defines a plane in space and the solution is a single point where all three planes intersect. • The intersection of any 2 planes is a line. • The 3 planes can intersect in such a way that the intersection is a line (dependent system) or they can meet only 2 at any given spot (or be parallel) (inconsistent system)

  22. Solving symbolically/analytically • Use one equation to remove one variable from the other 2 equations • Solve the resulting two variable system • Return and find the value of the third variable • To remove the first variable the techniques are the same as for two variable systems.

  23. Example: by substitution • x + y + z = 7 (eq1) 2x – 3y + z = -62 (eq2) 2x + 4y + 3z = 42 (eq3) • Solution (-5,16,-4)

  24. Example: solve by elimination • 2x – 3y + 5z = -8 (eq1) • 3x + 3y – 2z = 9 (eq2) • 5x – y + z = 8 (eq3) • x = 2 y = -1 z=-3

  25. Inconsistent: analytically • If at any time all variables cancel and the equation is false the system is inconsistent • Example: • 5x + 9y + z = 9 3x + 4y + z = 12 4x + 3y +2z = 12

  26. Dependent - analytically • If at any time the variables disappear and the resulting equation is true you have a dependent system • Example: • 6x – 4y + z = 9 8x – 3y + z = 7 4x – 5y + z = 11

  27. More Examples 2x – 3y +2z = -7 x + 4y – z = 10 3x + 2y + z = 4 2x + 5y – z = 9 x - y + 2z = -1 x + 3y - z = 6 2x + y + 3z = 2 x - 11y + 4z = -54 -5x + 8y – 12 z = 43 2x + y = - 3 2y + 16z = -18 -7x – 3y + 4z = 6

  28. Linear inequalities Chapter 2 – Section 3

  29. 2 variable inequalities • The solution set for a one variable inequality is an interval - it has a start and a stop and can be described easily • The solution set of a two variable inequality is a section of the plane which can only be described in the most abstract sense therefore graphs on linear inequalities are very helpful

  30. Graphing Linear inequalities • Graph line. • Determine the direction of shading either by analyzing the equation or substituting a point • Shade the section that is true for the inequality [check a point to verify] • Most graphing utilities have a way to do this on them

  31. Example • y > 2x – 7

  32. Example • 2x + 5y < 10

  33. Example

  34. Solutions/test point • a given point is a solution to the problem if it lies in the solution area • a given point is a solution if it makes the algebraic sentence true

  35. Example: • Mark need to buy notebooks and pencils. The notebooks cost $4.50 each and the pencils packages cost $2.75 each. He can spend no more than $50. • write an algebraic inequality to model this. • graph the inequality • can he buy 7 notebooks and 10 packages of pencils

  36. Systems of linear inequalities Chapter 2 – section 4

  37. Systems of inequalities • A problem that involves two or more conditions on the domain and range is a system • Like single inequalities (only worse) a graph is the best way to see the solution set for this type of problem • the solution set is the intersection of the two inequalities – if it exists it is one section of the plane

  38. Systems of inequalities • Graph both lines • Determine the section which makes BOTH inequalities true and shade ONLY that section • You can use test points or analyze the statement

  39. Inequality Systems - examples Y < 3 Y ≥ 2x - 4

  40. Inequality Systems - examples 3x + 2y < 8 3y – 2x < 15

  41. Example • x+ y > 2 x – y > 5

  42. Inequality Systems - examples y< 2x -3 y> 4 – 2x x< 4

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