1. Unit 4 Seminar Agenda Slope  What it is, What it looks like, how to find it  Ordered Pairs Types of Lines  Diagonal, Horizontal, and Vertical  Parallel and Perpendicular Finding the Equation of a Line Solving Systems of Equations  Graphing  Substitution  Addition/Elimination  Consistent/Inconsistent, Dependent/Independent

2. The SLOPE of a Line How STEEP a line is How DIAGONAL a line is How big of an angle a line makes with the x – axis A comparison of the vertical change a line makes with the horizontal change a line makes.

3. The SLOPE of a Line Will be POSITIVE if the diagonal line is heading uphill (increasing) as you look at it from left to right From top to bottom y = 1.5x - 6 y = (13/2)x + 3/2 y = (1/10)x + 4

4. The SLOPE of a Line Will be NEGATIVE if the diagonal line is heading downhill (decreasing) as you look at it from left to right From top to bottom y = -1.5x - 6 y = (-13/2)x + 3/2 y = (-1/10)x + 4

5. The SLOPE of a Line Will be ZERO if the line is a HORIZONTAL line …. From top to bottom y = 5 y = 5/2 y = -1.2 y = -7 NEVER cross the x – axis They have NO x – intercept Are of the form y = b

6. The SLOPE of a Line Will be UNDEFINED if the line is a VERTICAL line …. NEVER cross the y – axis They have NO y – intercept Are of the form x = a From left to right x = -15/2 x = -3 x = 2.2 x = 6

7. The SLOPE of a Line To calculate the slope of a line algebraically This IS VERY EASY ARITHMETIC … LET IT BE EASY!

8. Find the slope of the line through the points (2, 3) and (-4, 2) Substitute the given values into the formula Perform the arithmetic (reduce fractions if possible) Remember … the slope formula is :

9. Find the slope of the line in the given graph Remember … the slope formula is: Substitute the values into the formula Perform the arithmetic (reduce fractions if possible)

10. Parallel Lines NEVER intersect. If two lines are parallel, Their slopes are EXACTLY EQUAL Two lines with equal slopes are parallel.

11. Perpendicular Lines Intersect once. If two lines are perpendicular, Their slopes are NEGATIVE RECIPROCALS Two lines with negative reciprocal slopes are perpendicular.

12. Equations of a Lines The most useful equation of a line is the slope – intercept equation. y = mx + b m is the symbol for slope b is the symbol for the y – intercept Given ANY linear equation, if we rearrange it so there are NO grouping symbols, all like terms are combined, AND one side as the y all alone … then we are in slope intercept form.

13. SLOPE – INTERCEPT Equation y = mx + b … once you get your equation in this form, then you KNOW the slope and you KNOW the y – intercept (no calculations required to find them!) For example: y = 6x – 5 SLOPE = 6 Y – INTERCEPT (0, -5)

14. SLOPE – INTERCEPT Equation This also means if you know the slope and you know the y – intercept, then you can come up with the equation very easily. Given slope = -1/4 and (0, 1/3), find the equation in slope – intercept form. y = mx + b …. so ….

15. To find the slope – intercept form of data (if the slope and y intercept are not just given to you) Find the SLOPE  Use the formula (if given ordered pairs or the graph)  Use the definition of parallel or perpendicular line

16. To find the slope – intercept form of data (if the slope and y intercept are not just given to you) Find the SLOPE  Use the formula (if given ordered pairs or the graph)  Use the definition of parallel or perpendicular line Find the EQUATION  Choose one of the given ordered pairs  Substitute the pair you choose and the slope you found into the formula  If you only have one ordered pair, then use it  If you have two pairs, choose one of them (it does not matter which you choose!)

17. Find the slope – intercept form of the equation passing through (-2, -2) and perpendicular to -5x + y = 4 Find the SLOPE: Not given directly to use … BUT … we are told, our line is perpendicular to -5x + y = 4. Let’s find the slope of this line, then use the definition of perpendicular lines to find our slope.  -5x + y = 4  y = 5x + 4 … slope of this line is 5 … our line is perpendicular, so our line has a slope of -1/5

18. Find the slope – intercept form of the equation passing through (-2, -2) and perpendicular to -5x + y = 4 Find the SLOPE: -1/5 Find the EQUATION:  Choose one of the given ordered pairs  Substitute the pair you choose and the slope you found into the formula

19. Systems of Equations Definition: A system of equations  Two or more equations  With two or more unknowns  That may or may not contain a common solution.

20. Systems of Equations Solutions to a system of equations  A system of equations with TWO equations can have ONE solution, NO solutions, or INFINITE solutions.  The solution to a system of equations is an ordered pair that satisfies ALL the equations of the system.

21. Methods Used to Solve Systems of Equations There are many different methods available. Regardless of the method you use, you will get the same answer. Guessing Graphing Substitution Addition/Elimination

22. Solving Systems of Equations The GUESSING Method Ineffective and inefficient …. Will not be discussed any more

23. Solving Systems of Equations The GRAPHING Method Graph the equations on the same graph (using the techniques from the previous unit)

24. Solving Systems of Equations The GRAPHING Method Now think about this: if you graph two straight lines on the same grid, three possible pictures emerge: 1. The two lines INTERSECT They have ONE point in common There is ONE unique solution to the system, in the form (x, y)

25. Solving Systems of Equations The GRAPHING Method Now think about this: if you graph two straight lines on the same grid, three possible pictures emerge: 2. The two lines are PARALLEL to each other They have NO points in common There is NO SOLUTION to the system

26. Solving Systems of Equations The GRAPHING Method Now think about this: if you graph two straight lines on the same grid, three possible pictures emerge: 3. The two lines are the SAME— called COINCIDENTAL lines They have ALL points in common There are INFINITELY MANY SOLUTIONS to the system

27. Solving Systems of Equations The SUBSTITUTION Method Rearrange one of the equations (it does not matter which one) to get the X or the Y all alone (it does not matter which one). Take the result and substitute it into the OTHER equation. Solve for the unknown in this new equation. Substitute your result into one of the ORIGINAL equations and solve for the unknown. Write your solution as an ordered pair.

28. Solving Systems of Equations The SUBSTITUTION Method Rearrange one of the equations (it does not matter which one) to get the X or the Y all alone (it does not matter which one).

29. Solving Systems of Equations The SUBSTITUTION Method Take the result and substitute it into the OTHER equation.

30. Solving Systems of Equations The SUBSTITUTION Method Solve for the unknown in this new equation.

31. Solving Systems of Equations The SUBSTITUTION Method Substitute your result into one of the ORIGINAL equations and solve for the unknown.

32. Solving Systems of Equations The SUBSTITUTION Method Write your solution as an ordered pair.

33. Solving Systems of Equations The ADDITION/ELIMINATION Method While the substitution method allowed us to make an equation have only one variable by replacement, the elimination method allows us to do the same thing by actually getting rid of one variable (temporarily, of course).

34. Solving Systems of Equations The ADDITION/ELIMINATION Method The goal in this method is to get the numbers in front of both x’s OR both y’s to be additive inverses of one another (same number, opposite signs).

35. Solving Systems of Equations The ADDITION/ELIMINATION Method The goal in this method is to get the numbers in front of both x’s OR both y’s to be additive inverses of one another (same number, opposite signs). That way … when we ADD the equations together, one of the variables will be ELIMINATED.

36. Solving Systems of Equations The ADDITION/ELIMINATION Method Make sure BOTH equations are in Ax + By = C form

37. Solving Systems of Equations The ADDITION/ELIMINATION Method Make sure BOTH equations are in Ax + By = C form Find (because they already exist) or create (by multiplying) additive inverses of one variable. IT DOES NOT MATTER WHICH VARIBLE YOU ELIMINATE

38. Solving Systems of Equations The ADDITION/ELIMINATION Method Make sure BOTH equations are in Ax + By = C form Find (because they already exist) or create (by multiplying) additive inverses of one variable. Add the equations together to eliminate one of the variables.

39. Solving Systems of Equations The ADDITION/ELIMINATION Method Make sure BOTH equations are in Ax + By = C form Find (because they already exist) or create (by multiplying) additive inverses of one variable. Add the equations together to eliminate one of the variables. Solve for the unknown in this new equation.

40. Solving Systems of Equations The ADDITION/ELIMINATION Method Make sure BOTH equations are in Ax + By = C form Find (because they already exist) or create (by multiplying) additive inverses of one variable. Add the equations together to eliminate one of the variables. Solve for the unknown in this new equation. Substitute your result into one of the ORIGINAL equations and solve for the unknown.

41. Solving Systems of Equations The ADDITION/ELIMINATION Method Make sure BOTH equations are in Ax + By = C form Find (because they already exist) or create (by multiplying) additive inverses of one variable. Add the equations together to eliminate one of the variables. Solve for the unknown in this new equation. Substitute your result into one of the ORIGINAL equations and solve for the unknown. Write your solution as an ordered pair

42. Solving Systems of Equations The ADDITION/ELIMINATION Method Make sure BOTH equations are in Ax + By = C form Find (because they already exist) or create (by multiplying) additive inverses of one variable. (It does not matter which variable you eliminate!) Add the equations together to eliminate one of the variables. Solve for the unknown in this new equation. Substitute your result into one of the ORIGINAL equations and solve for the unknown. Write your solution as an ordered pair

43. Vocabulary of Systems of Equations  Some other terminology comes into play when you’re dealing with systems of equations.

44. Vocabulary of Systems of Equations  Some other terminology comes into play when you’re dealing with systems of equations.  DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).

45. Vocabulary of Systems of Equations  Some other terminology comes into play when you’re dealing with systems of equations.  DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).  INDEPENDENT: the graph of two lines shows up as two lines (parallel and intersecting lines both qualify).

46. Vocabulary of Systems of Equations  Some other terminology comes into play when you’re dealing with systems of equations.  DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).  INDEPENDENT: the graph of two lines shows up as two lines (parallel and intersecting lines both qualify).  CONSISTENT: there is at least one solution to the system (intersecting lines have one and coincidental lines have infinitely many).

47. Vocabulary of Systems of Equations  Some other terminology comes into play when you’re dealing with systems of equations.  DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).  INDEPENDENT: the graph of two lines shows up as two lines (parallel and intersecting lines both qualify).  CONSISTENT: there is at least one solution to the system (intersecting lines have one and coincidental lines have infinitely many).  INCONSISTENT: there is no solution to the system (parallel lines).

48. Vocabulary of Systems of Equations  Some other terminology comes into play when you’re dealing with systems of equations.  DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).  INDEPENDENT: the graph of two lines shows up as two lines (parallel and intersecting lines both qualify).  CONSISTENT: there is at least one solution to the system (intersecting lines have one and coincidental lines have infinitely many).  INCONSISTENT: there is no solution to the system (parallel lines).

49. Systems of Equations Solutions to a system of equations  ONE SOLUTION: Independent, Consistent  NO SOLUTIONS: Independent, Inconsistent  INFINITE SOLUTIONS: Dependent, Consistent