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Unit 4 Seminar Agenda 4.1 Variation 4.2 Linear Inequalities 4.3 Graphing Linear Equations

Filling your tank is direct variation I purchased 15 gallons of gasoline for $ How much would 20 gallons have cost? Cost = price per gallon times # gallons C = kg ‘general formula’ $58.35 = k (15)‘fill-in all but ‘k’’ $58.35 = 15k‘solve for k’ k = $3.89 per gallon C = $3.89 (20) = $77.80 for 20 gallons‘use k and the formula to answer’ 3

4 Direct Variation If a variable y varies directly with a variable x, then y = kx, where k is the constant of proportionality. 4

5 If R varies directly L, and L is 30 when R is 0.24, then determine R when L is 40. Real World Application: Finding the length of wire or spring, L, based resistance, R. R = kL 0.24 = k(30) 0.24/30 = k = k R = 0.008L R = 0.008(40) R =

6 Inverse Variation If a variable y varies inversely with a variable x, then y = k/x, where k is the constant of proportionality. 6

7 If t is inversely proportional T and t is 2 when T is 75, then find t when T is 80. Real World Application: Finding the time it will take to melt, t, based on temperature, T. t = k/T 2 = k/ = k t = 150/T t = 150/80 t = minutes 7

Solving Inequalities: Use the same steps as solving equations. Except: If you multiply or divide both sides by the same negative number, then you must ‘flip’ the symbol. Why??? 3 < 73 < 7 3(2) < 7(2)3(-2) < 7(-2) 6 < 14-6 < -14…..NO! Makes sense!-6 > -14 8

9 4x + 9 > 25 4x > x > 16 x > 4 9

x ≤ x - 10 ≤ x ≤ 11 -3x/-3 ≥ 11 / -3 x ≥ -11 / 3 10

11 56 > -9x + 2 > > -9x > > -9x > 20 54/-9 < -9x/-9 < 20/-9 -6 < x < -2 2/9 11

12 Coordinate Grid x axis y axis origin Quadrant IQuadrant II Quadrant III Quadrant IV A B C D

13 Find three points on the line 2x – 3y = 12 First point: I will replace x with 0. 2(0) – 3y = 12 0 – 3y = 12 -3y = 12 -3y/(-3) = 12/(-3) y = -4 Ordered pair is (0, -4) 13

14 Find three points on the line 2x – 3y = 12 Second point: I will replace x with 2. 2(2) – 3y = 12 4 – 3y = 12 4 – 3y – 4 = 12 – 4 -3y = 8 -3y/(-3) = 8/(-3) y = -8/3 Ordered pair is (2, -8/3) 14

15 Find three points on the line 2x – 3y = 12 Third point: I will replace x with -3. 2(-3) – 3y = – 3y = – 3y + 6 = y = 18 -3y/(-3) = 18/(-3) y = -6 Ordered pair is (-3, -6) 15

16 2x – 3y = 12(0,-4)(2, -8/3)(-3,-6) 16

Positive Slope As x values increase, y values also increase

Negative Slope As x values increase, y values decrease

Horizontal Line Form: y = constant number

Vertical Line Form: x = constant number

Slope Slope measures the steepness of a line. You can use the formula: You can use counting: m = rise/run 21

Finding slope given 2 points m = 2/3 Find the slope between (1,3) and (4,5)

Example Find the slope of the line that passes through the points (6, 4) and (6, 2). m = UNDEFINED (Division by zero is undefined)

The Slope-Intercept Formula y = mx + b m represents the slope (how steep it is) b represents the y intercept (where it crosses the y axis 24

Finding slope and y-intercept given an equation First solve for y. When you have it in the form y = mx + b then m is your slope and b is your y-intercept. 3x + y = 7 y = -3x + 7 m = -3 is your slope b = (0, 7) is your y-intercept

Graph y = -3x + 7

Write the equation of the line shown.