3.7 – variation

Direct Variation: y varies directly as x (y is directly proportional to x), if there is a nonzero constant k such that

3.7 – Variation

The number k is called the constant of variation or the constant of proportionality

.kxy

Verbal Phrase Expression𝑦 𝑣𝑎𝑟𝑖𝑒𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 h𝑤𝑖𝑡 𝑥 𝑦=𝑘𝑥

𝑠𝑣𝑎𝑟𝑖𝑒𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 h𝑤𝑖𝑡 h𝑡 𝑒 𝑠𝑞𝑢𝑎𝑟𝑒𝑜𝑓 𝑡 𝑠=𝑘𝑡 2

𝑦 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝 . h𝑤𝑖𝑡 h𝑡 𝑒𝑐𝑢𝑏𝑒𝑜𝑓 𝑧 𝑦=𝑘 𝑧3

𝑢𝑖𝑠𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝 . h𝑤𝑖𝑡 h𝑡 𝑒𝑠𝑞 .𝑟𝑡 .𝑜𝑓 𝑣 𝑢=𝑘√𝑣

Direct Variation

kxy 824 k

k824

Suppose y varies directly as x. If y is 24 when x is 8, find the constant of variation (k) and the direct variation equation.

3k

xy 3direct variation equation

constant of variation

xy

39

515

927

1339

3.7 – Variation

kwd 567 k

k567

Hooke’s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 56-pound weight stretches a spring 7 inches, find the distance that an 85-pound weight stretches the spring. Round to tenths.

81

k

xy31

direct variation equation


8531

y

6.10y inches

3.7 – Variation

Inverse Variation: y varies inversely as x (y is inversely proportional to x), if there is a nonzero constant k such that

The number k is called the constant of variation or the constant of proportionality.

.xky

Verbal Phrase Expression𝑦 𝑖𝑠𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 h𝑤𝑖𝑡 𝑥 𝑦=

𝑘𝑥

𝑠𝑣𝑎𝑟𝑖𝑒𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 h𝑤𝑖𝑡 h𝑡 𝑒𝑠𝑞𝑢𝑎𝑟𝑒𝑜𝑓 𝑡 𝑠= 𝑘𝑡 2

𝑦 𝑖𝑠𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑧 4 𝑦=𝑘𝑧4

𝑢𝑣𝑎𝑟𝑖𝑒𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 h𝑤𝑖𝑡 h𝑡 𝑒𝑐𝑢𝑏𝑒 .𝑟𝑡 .𝑜𝑓 𝑣 𝑢=𝑘

3√𝑣

3.7 – Variation

Inverse Variation

xk

y

36

k

k18

Suppose y varies inversely as x. If y is 6 when x is 3, find the constant of variation (k) and the inverse variation equation.

xy

18



xy

36

92

101.8

181

3.7 – Variation

tk

r

430

k

k120

The speed r at which one needs to drive in order to travel a constant distance is inversely proportional to the time t. A fixed distance can be driven in 4 hours at a rate of 30 mph. Find the rate needed to drive the same distance in 5 hours.

xr

120



5120

r

24r mph

3.7 – Variation

Joint VariationIf the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional, to the other variables.

Verbal Phrase Expression

𝑦 𝑣𝑎𝑟𝑖𝑒𝑠 𝑗𝑜𝑖𝑛𝑡𝑙𝑦 h𝑤𝑖𝑡 𝑥 𝑎𝑛𝑑 𝑧 𝑦=𝑘𝑥𝑧

𝑧=𝑘𝑟 𝑡2

𝑉=𝑘𝑇𝑃

𝐹=𝑘𝑚𝑛𝑟 2

3.7 – Variation

Joint Variation

𝑧=𝑘𝑥𝑦

z varies jointly as x and y. x = 3 and y = 2 when z = 12. Find z when x = 4 and y = 5.

12=𝑘 (3 ) (2 )2=𝑘𝑧=2 𝑥𝑦𝑧=2 (4 ) (5 )𝑧=40

3.7 – Variation

Joint Variation

𝑉=𝑘h𝑟 2

V varies jointly as h and . V = 402.12 cubic inches, h = 8 inches and r = 4 inches. Find V when h = 10 and r = 2.

402.12=𝑘 (8 ) ( 4 )2 3.142=𝑘𝑉=3.142h𝑟 2 𝑖𝑛3𝑉=125.68

The volume of a can varies jointly as the height of the can and the square of its radius. A can with an 8 inch height and 4 inch radius has a volume of 402.12 cubic inches. What is the volume of a can that has a 2 inch radius and a 10 inch height?

𝑉=3.142 (10 )22

3.7 – Variation

3 3 02

x yx y

A system of linear equations allows the relationship between two or more linear equations to be compared and analyzed.

02 10x yx y

7 14

y xy

7 292 432

y x

y x

y x

4.1 - Systems of Linear Equations in Two Variables

Determine whether (3, 9) is a solution of the following system. 5 2 3

3x yy x

5 3 2 39

15 18 3 3 3

9 33

9 9

Both statements are true, therefore (3, 9) is a solution to the given system of linear equations.


Determine whether (3, -2) is a solution of the following system.

2 83 4

x yx y

2 3 2 8

6 2 8 8 8

3 23 4

3 6 4

Both statements are not true, therefore (3, -2) is not a solution to the given system of linear equations.

3 4


Solving Systems of Linear Equations by Graphing

4 13

y xy

: 1,3Solution

3 3 3 4 1 1

3 3


Solving Systems of Linear Equations by Graphing

2 03x y

x y

: 1, 2Solution

3 3

2 1 2 0

0 0

3y x

1 2 3

2y x


Solving Systems of Linear Equations by the Addition Method

9357 3328

126315

2036610 2950627

9141217


(Also referred to as the Elimination Method)

6413

yxyx

13 yx

113 y

64 yx

13 yx

13 y2 y

2y

2,1Solution

77 x

77 x1x

4.1 - Systems of Linear Equations in Two VariablesSolving Systems of Linear Equations by the Addition Method


3210145

yxyx

145 yx

14515

y

32102 yx

145 yx

141 y24 y

21,

51

Solution525 x 5

1x

145 yx6420 yx 2

1y



943652

yxyx

6523 yx 6052 x

9432 yx

652 yx

62 x3x

0,3Solution

07 y0y

18156 yx1886 yx



201481074

yxyx

10742 yx20148 yx

00

20148 yx20148 yx

True Statement


Solution: All realsLines are the same




23593

yxyx

233 yx593 yx

10

593 yx693 yx

lines are parallel

False Statement

No Solution



Solving Systems of Linear Equations by Substitution

5 2 33

x yy x

325 yx 3325 xx

3x

365 xx3 x

xy 3

33y

9y

9,3Solution


Solving Systems of Linear Equations by Substitution

2 83 4

x yx y

43 yx43 yx

0y

82 yx

07 y

82 yx802 x

4x

0,4Solution

8432 yy886 yy

887 y82 x


Example4.1 - Systems of Linear Equations in Two Variables

LCD: 6

LCD: 15

Solution(2 ,−1)

3.7 – variation

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