2. Relations Math 314

3. Time Frame Calculations Slope Point Slope Parameters Word Problems

4. Calculations Sample Questions Equations – Find x 1 + 1 = 1 2 3 x Find LCD = 6x 3x + 2x = 6 5x = 6 x = 6/5 x = 1.2

5. Calculations Half a number plus a third of the same number is 11. What is the number. Let x = the information you know the least about. 1x + 1x = 11 2 3 3x + 2x = 66 5x = 66 x = 66/5 x = 13.2

6. A Difficult Word Problem Three people went bowling who are Sarah (S), Lucas (L) & Corey (C). Sarah bowled twice as better than Lucas. Corey bowled 15 points less than three times as better than Lucas. All together their scores were 165. How much did each person bowl.

7. Solution Sarah: 2x Lucas x Corey 3x – 15 2x + x + 3x – 15 = 165 6x – 15 = 165 6x = 180 x = 30 One more step! Lucas  30; Sarah  60 & Corey  75 points.

8. Distance Between Numbers Which is closer to 11/24; ½ or 1/3? Put on a number line 1/3 11/24 ½ To find the distance… subtract (highest from the lowest)

9. Distance Between Numbers 11 – 1 24 3 = 11 – 8 24 = 3 24 1 – 11 2 24 = 12 – 11 24 = 1 24 Which one is closer? 1/24

10. Substitution Sometimes we look at a relationship as a formula Consider 2x + 8y = 16 We have moved away from a single variable equation to a double variable equation It cannot be solved as is!

11. Substitution If we know x = 4 2x + 8y = 16 2(4) + 8y = 16 8 + 8y = 16 8y = 8 y = 1

12. Substitution We could say that the point x = 4 and y = 1 or (4,1) satisfies the relationship. Ex #2. Given the relationship 5x – 7y = 210, use proper substitution to find the coordinate (2,y) (2,y)  5x – 7y = 210 5(2) – 7y = 210 10 – 7y = 210 -7y = 200 y = - 28.57 (2, -28.57)

13. Substitution Ex. #3: Given the relationship 8x + 5y = 80 (x,8) (x,8)  8x + 5y = 80 8x + 5(8) = 80 8x + 40 = 80 8x = 40 x = 5 (5,8)

14. Substitution Ex: #4 Given the relationship y= 3x 2 – 5x – 2 (-3,y) (-3,y)  y = 3 (-3) 2 – 5 (-3) – 2 y = 3 (9) + 15 – 2 y = 40 (-3,40) Stencil #2 (a-j)

15. Substitution Given the relationship

16. Linear Relations We recall… Zero constant relation – horizontal Direct relation – through origin Partial relation – not through origin The characteristic here is the concept of a straight line – a never changing start and where it crosses the y axis

17. Example Line A Line B We say line A has a more of a slant slope or a steeper slope than Line B (a of 6 compared to 2 is steeper or slope of -6 compared to -2 is steeper).

18. Variation Relations Name of RelationFormulaGraph Direct Relationy = mx Partial Relationy = mx + b Zero Variationy = b Inverse Variationy = m x

19. Slope What makes a slope? Rise Run We define the slope as the ratio between the rise and the run Slope = m = rise run

20. Formula for Slope If we have two points (x 1, y 1 ) (x 2, y 2 ) Slope = m = y 1 – y 2 = y 2 – y 1 x 1 – x 2 x 2 – x 1 Remember it is Y over X! Maintain order

21. A (x 1, y 1 ) B(x 2, y 2 ) Slope Consider two points A (5,4), B (2, 1) what is the slope?

22. Calculating Slope Slope = m = y 1 – y 2 = y 2 – y 1 x 1 – x 2 x 2 – x 1 (5, 4) (2, 1) 4 - 1 5 - 2 3 3 m = 1 (x 1,y 1 ) (x 2,y 2 )

23. Ex # 2 A = (-4, 2) B=(2, -4) (x 1,y 1 ) (x 2,y 2 ) -4 – 2 2 - - 4 -6-6 6 m = -1

24. (4, 5) (1, 1) Ex #3 (x 2,y 2 ) (x 1,y 1 )

25. Understanding the Slope If m or the slope is 2 this means a rise of 2 and a run of 1 (2 can be written as 2 ) 1 If m = - 5, this means a rise of -5 and right 1 If m= -2 this means rise of -2 right 3 3 Rise can go up or down, run must go right

26. Consider y = 2x + 3 What is the slope, y intercept, rise & run? We can write the slope 2 as a fraction 2 1 We have a y intercept of 3 This means rise of 2, run of 1 Look at previous slide for slope of 4/3

27. Ex#1: y=2x+3 0,3 (1,5) Question: Draw this line What is the y intercept? What is the slope What does the slope mean? Where can you plot the y intercept? Up 2, Right 1

28. (-4, 2) (2,2) If a line//x-axis slope = 0 Example What do you think the slope will be; calculate it.

29. (2,-3) (2,2) If a line // y-axis: slope is undefined Example zero!

30. In Search of the Equation We have seen that the linear relation or function is defined by two main characteristics or parameters A parameter are characteristics or how we describe something If we consider humans, a parameter would be gender. (We have males & females). There can be many other parameters (blonde hair, blue eyes, etc.)

31. In Search of the Equation Notes The parameters we are concerned with are… Slope = m = the slope of the line y intercept = b = where the line touches or crosses the y axis (It can always be found by replacing x = 0) x intercept = where on the graph the line touches or crosses the x axis. (let y = 0)

32. In Search of the Equation Notes We stated in standard form the equation for all linear functions by y = mx + b. Recall… y is the Dependent Variable (DV) m is the slope x is the Independent Variable (IV) b is the y intercept parameter The key is going to be finding the specific parameters.

33. General Form You will also be asked to write in general form General Form Ax + By + C = 0 A must be positive Maintain order x, y, number = 0 No fractions

34. General Form Practice Consider y = 6x – 56 -6x + y + 56 = 0 6x – y – 56 = 0

35. Standard & General Form Example #1 State the equation in standard and general form. Consider find the equation of the linear function with slope of m and passing through (x, y). m = -6 (-2, -3) (-2, -3)  -3 = -6 (-2) + b -3 = 12 + b -15 = b b = -15

36. Example #1 Solution Con’t y = -6x – 15 (Standard) Now put this in general form 6x + y +15 = 0 (General)

37. Standard & General Form Ex. #2 m = -2 (5, - 3) 3 -3 = (-2) (5) + b 3 -3 = -10 + b 3 -9 = -10 + 3b 1 = 3b b = 1/3 y = -2 x + 1 (SF) 3 3 Now General form Get rid of the fractions; how? Given y = -2 x + 1 3 3… Anything times the bottom gives you the top 3y = -2x + 1 2x + 3y – 1 = 0

38. Standard and General Form Ex #3 m = 4 5 (-1, -1) -1 = 4 x + b 5 -5y = -4x + 5b  5 (-1) = 4 (-1) + 5b -5 = -4 + 5b -1 = 5b b = -1/5 y = 4x – 1 5 5 5 x – 1/5 (standard form) 5y = 4x – 1 -4x +5y + 1 = 0 4x – 5y – 1 = 0 (general form)

39. The Point Slope Method Con’t Consider, find the equation of the linear function with slope 6 and passing through (9 – 2). Take a look at what we know based on this question. m = 6 x = 9 y = -2

40. Finding the Equation in Standard Form We know y = mx + b We already know y = 6x + b What we do not know is the b parameter or the y intercept We will substitute the point (9, -2)  - 2 = (6) (9) + b -2 = 54 + b -56 = b b = - 56 y = 6x – 56 (this is Standard Form) Standard from is always y = mx + b (the + b part can be negative… ). You must have the y = on the left hand sides and everything else on the right hand side.

41. General Form In standard form y = 6x – 56 In general form -6x + y + 56 = 0 6x – y – 56 = 0

42. Example #1 8a on Stencil In the following situations, identify the dependent and independent variables and state the linear relations Little Billy rents a car for five days and pays $287.98. Little Sally rents a car for 26 days and pays $1195.39. D.V  $ Money $ I.V.  # of days

43. Example #1 Soln Con’t Try and figure out the equation y = mx + b (you want 1 unknown) (5, 287.98) (26, 1195.39) m = (287.98 – 1195.39) 5 – 26 m = 43.21 Unknown

44. Example #1 Soln Con’t Solve for b… y = mx + b (5, 287.98)  287.98 = 43.21 (5) + b 287.98 = 216.05 + b 71.93 = b b = 71.93 y = 43.21x + 71.93

45. Example #2 8 b on Stencil A company charges $62.25 per day plus a fixed cost to rent equipment. Little Billy pays $1264.92 for 19 days. I.V. # of days D.V. Money m = 62.25

46. Example #2 8a Soln y = mx + b (19, 1264.92)  1264.92 = 62.25 (19) + b 1264.92 = 1182.75 + b 82.17 = b b = 82.17 y = 62.25x + 82.17

47. Solutions 8 c, d, e 8c) IV # of days; DV $ y = 47.15x + 97.79 8d) IV # of days; DV $ y = 89.97x + 35.22 8e) IV # of days DV $ y= 45.13x + 92.16

48. Homework Help What is the value of x given 3 = 1 + 1 4 2 x Eventually, x on the left side, number on the right side 3 – 1 = 1 4 2 x 6x – 4x = 8 -2x = 8 x = -4 Important step to understand

49. Homework Help What is the opposite of ½ ? Answer is – ½ If asked what is the opposite of subtracting two fractions… i.e. ¼ - ½, find the answer (lowest common denominator and then reverse the sign. When told price increases 10% each year… calculate new price after year 1 and then multiply that number by.1 again to calculate price increase for year 2. For example, you have $100 and increases 10%. After year 1  $110 (100 x.1 + 100) & after year two $121 (110 x.1 + 110).