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Linear Relations with Substitution and Slope

Explore solving equations with substitution method and analyzing slope in linear relations. Learn to find coordinates using substitution. Understand the concept of slope and its importance in relation to straight lines.

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Linear Relations with Substitution and Slope

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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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