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

3 Direct Variation If a variable y varies directly with a variable x, then y = kx, where k is the constant of proportionality. 3

4 The resistance, R, of a wire varies directly as its length, L. If the resistance of a 30 ft length of wire is 0.24 ohm, determine the resistance of a 40 ft length of wire? R = kL 0.24 = k(30) 0.24/30 = k = k Now that k is determined, we can find R if L=40: R = 0.008L R = 0.008(40) R = 0.32 ohm 4

5 Inverse Variation If a variable y varies inversely with a variable x, then y = k/x, where k is the constant of proportionality. 5

6 The time, t, for an ice cube to melt is inversely proportional to the temperature, T, of the water in which the ice cube is placed. If it takes an ice cube 2 minutes to melt in 75 degree F water, how long will it take an ice cube of the same size to melt in 80 degree F water? t = k/T 2 = k/ = k Now that k is determined, in the second part of the problem, we are now looking for t when T=80. t = 150/T t = 150/80 t = minutes Always check if the answer is “reasonable”. 6

7 Joint Variation The general form of a joint variation, where y varies directly as x and z, is y = kxz, where k is the constant of proportionality. 7

8 The volume, V, of a pyramid varies jointly as the area of its base, B, and height, h. If the volume of a pyramid is 12 cubic feet when the area of the base is 4 square feet and the height is 9 feet, find the volume of a pyramid when the area of the base is 16 square feet and the height is 9 feet. V = kBh 12 = k(4)(9) 12 = 36k 12/36 = k 1/3 = k V = (1/3)Bh V = (1/3)(16)(9) V = 48 cubic feet 8

9 EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

10 C = k/J 10 EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

11 C = k/J 7 = k/ = k 11 EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

12 C = k/J 7 = k/ = k Now that we know k C = k/J C = 4.9/12 C = EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

13 Inequality signs < is less than ex: -3 < 5 ≤ is less than or equal to ex: x ≤ 0 > is greater than ex: 5 > 1 ≥ is greater than or equal to ex: x ≥ 9 13

14 4x + 9 > 25 Use the same principles you learned with equations (except for one caveat). 4x > x > 16 x > 4 14

x ≤ x - 10 ≤ x ≤ 11 -3x/-3 ≥ 11/-3 (this is the exception: notice the change in direction of the inequality) x ≥ -11/3 15

16 -(1/4)x - 5 > 9 -(1/4)x > (1/4)x > 14 (-4)(-1/4)x < (-4)(14) (note change in inequality sign) x <

17 56 > -9x + 2 > > -9x > > -9x > 27 54/-9 < -9x/-9 < 27/-9 (sign change again!) -6 x > -3 is not true 17

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

19 Example: x + y = 5. x = 0 and y = 5 gives ordered pair (0,5) x = 1 and y = 4 gives ordered pair (1, 4) x = 2 and y = 3 gives ordered pair (2, 3) x = 3 and y = 2 gives ordered pair (3,2) x = 4 and y = 1 gives ordered pair (4, 1) x = 5 and y = 0 gives ordered pair (5, 0)

20 Graphing Linear Equations x + y = 5

21 Example: 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) 21

22 Example: 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) 22

23 Example: 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) 23

24 (0, -4) (2, -8/3) (-3, -6) 24 Graph of 2x -3y = 12

25 4x + y = -1

26 Example: What are the x- and y- intercepts of 3x – 7y = 21? To find the x- intercept, let y = 0. 3x – 7(0) = 21 3x – 0 = 21 3x = 21 x = 7 (7,0) To find the y-intercept, let x = 0. 3(0) – 7y = 21 0 – 7y = 21 -7y = 21 y = -3 (0, -3)

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

Finding Slope Graphically (-6, -1) (2, 4) Up 5 Right 8 Slope is 5/8

Finding Slope Graphically (-4, 1) (3, -3) Right 7 Down 4 Slope is -4/7

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

Example: Find the slope of the line that passes through the points (-2, 6) and (-1, -2).

EVERYONE: Find the slope of the line that passes through the points (0, 3) and (7, -2)

m = -2 – 3 7 – 0 m = -5 7

Example Find the slope of the line that passes through the points (1, 7) and (-3, 7). m = 0

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

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

y = -3x + 7

Example First move your 2x to the other side simplify We need to isolate the y so divide both sides by 4 simplify