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Relations

Relations. Math 314. Time Frame. Slope Point Slope Parameters Word Problems. 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!. Substitution.

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Relations

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  1. Relations Math 314

  2. Time Frame • Slope • Point Slope • Parameters • Word Problems

  3. 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!

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

  5. 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)

  6. 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)

  7. Substitution • Ex: #4 Given the relationship y= 3x2 – 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) 

  8. Substitution • Given the relationship

  9. 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

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

  11. Variation Relations

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

  13. Formula for Slope • If we have two points (x1, y1) (x2, y2) • Slope = m = y1 – y2 = y2 – y1 x1 – x2 x2 – x1 • Remember it is Y over X! • Maintain order

  14. Slope A (x1, y1) B(x2, y2) • Consider two points A (5,4), B (2, 1) what is the slope?

  15. (x1,y1) (x2,y2) Calculating Slope • Slope = m = y1 – y2 = y2 – y1 x1 – x2 x2 – x1 (5, 4) (2, 1) 4 - 1 5 - 2 3 3 m = 1

  16. Ex # 2 A = (-4, 2) B=(2, -4) (x1,y1) (x2,y2) -4 – 2 2 - - 4 • 6 6 m = -1

  17. Ex #3 (4, 5) (x1,y1) (1, 1) (x2,y2)

  18. 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

  19. 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

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

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

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

  23. 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.)

  24. 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)

  25. 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.

  26. 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

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

  28. Standard & General FormExample #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

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

  30. 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 Standard & General Form Ex. #2

  31. 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) Standard and General Form Ex #3

  32. 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

  33. 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.

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

  35. 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

  36. 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 Unknown

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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).

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