3.7 – variation
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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
constant of variation
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
direct variation equation
constant of variation
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
direct variation equation
constant of variation
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.
4.1 - Systems of Linear Equations in Two Variables
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
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by Graphing
4 13
y xy
: 1,3Solution
3 3 3 4 1 1
3 3
4.1 - Systems of Linear Equations in Two Variables
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
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by the Addition Method
9357 3328
126315
2036610 2950627
9141217
4.1 - Systems of Linear Equations in Two Variables
(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
(Also referred to as the Elimination 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
4.1 - Systems of Linear Equations in Two VariablesSolving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
943652
yxyx
6523 yx 6052 x
9432 yx
652 yx
62 x3x
0,3Solution
07 y0y
18156 yx1886 yx
4.1 - Systems of Linear Equations in Two VariablesSolving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
201481074
yxyx
10742 yx20148 yx
00
20148 yx20148 yx
True Statement
4.1 - Systems of Linear Equations in Two Variables
Solution: All realsLines are the same
Solving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
4.1 - Systems of Linear Equations in Two Variables
23593
yxyx
233 yx593 yx
10
593 yx693 yx
lines are parallel
False Statement
No Solution
Solving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
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
4.1 - Systems of Linear Equations in Two Variables
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
4.1 - Systems of Linear Equations in Two Variables
Example4.1 - Systems of Linear Equations in Two Variables
LCD: 6
LCD: 15
Solution(2 ,−1)
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