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
y  kx.
The number k is called the constant of variation or the
constant of proportionality
Verbal Phrase
𝑦 𝑣𝑎𝑟𝑖𝑒𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑥
Expression
𝑦 = 𝑘𝑥
𝑠 𝑣𝑎𝑟𝑖𝑒𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑡
𝑠 = 𝑘𝑡 2
𝑦 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝. 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑐𝑢𝑏𝑒 𝑜𝑓 𝑧
𝑦 = 𝑘𝑧 3
𝑢 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝. 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑞. 𝑟𝑡. 𝑜𝑓 𝑣
𝑢=𝑘 𝑣
3.7 – Variation
Direct Variation
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.
y  kx
24  k 8
24
k
8
direct variation equation
y  3x
constant of variation
k 3
x
y
3
9
5
15
9
27
13
39
3.7 – Variation
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.
7  k 56
7
k
56
1
k
8
constant of variation
d  kw
direct variation equation
1
y x
3
1
y  85
3
y  10.6 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
k
y .
x
The number k is called the constant of variation or the
constant of proportionality.
Verbal Phrase
𝑦 𝑖𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑤𝑖𝑡ℎ 𝑥
𝑠 𝑣𝑎𝑟𝑖𝑒𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑡
𝑦 𝑖𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑧 4
𝑢 𝑣𝑎𝑟𝑖𝑒𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑐𝑢𝑏𝑒. 𝑟𝑡. 𝑜𝑓 𝑣
Expression
𝑦 = 𝑘𝑥
𝑠 = 𝑡𝑘2
𝑦 = 𝑧𝑘4
𝑢=
𝑘
3
𝑣
3.7 – Variation
Inverse Variation
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.
k
y
x
k
6
3
direct variation equation
18  k
18
y
x
constant of variation
x
y
3
6
9
2
10
18
1.8
1
3.7 – Variation
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.
k
r
t
k
30 
4
120  k
constant of variation
direct variation equation
120
r
x
120
r
5
r  24 mph
3.7 – Variation
Joint Variation
If 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 𝑤𝑖𝑡ℎ 𝑥 𝑎𝑛𝑑 𝑧
𝑦 = 𝑘𝑥𝑧
𝑧 𝑣𝑎𝑟𝑖𝑒𝑠 𝑗𝑜𝑖𝑛𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑟 𝑎𝑛𝑑
𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑡
𝑉 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑇 𝑎𝑛𝑑
𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑃
𝐹 𝑣𝑎𝑟𝑖𝑒𝑠 𝑗𝑜𝑖𝑛𝑡𝑙𝑦 𝑤𝑖𝑡ℎ 𝑚 𝑎𝑛𝑑 𝑛 𝑎𝑛𝑑
𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑟
𝑧 = 𝑘𝑟𝑡 2
𝑉 = 𝑘𝑇
𝑃
𝐹 = 𝑘𝑚𝑛
𝑟2
3.7 – Variation
Joint Variation
z varies jointly as x and y.
x = 3 and y = 2 when z = 12.
Find z when x = 4 and y = 5.
𝑧 = 𝑘𝑥𝑦
12 = 𝑘 3 2
2=𝑘
𝑧 = 2𝑥𝑦
𝑧=2 4 5
𝑧 = 40
3.7 – Variation
Joint Variation
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 varies jointly as h and 𝑟 2 .
V = 402.12 cubic inches, h = 8 inches and r = 4 inches.
Find V when h = 10 and r = 2.
𝑉 = 𝑘ℎ𝑟 2
402.12 = 𝑘 8 4
𝑉 = 3.142ℎ𝑟 2
2
3.142 = 𝑘
𝑉 = 3.142 10 22
𝑉 = 125.68 𝑖𝑛3
4.1 - Systems of Linear Equations in Two Variables
A system of linear equations allows the relationship between two or
more linear equations to be compared and analyzed.
3 x  3 y  0

 x  2y
 y  7 x 1

 y4
 x y 0

2 x  y  10
7

y

x

2

9

2

y   x  4
3

 y  2x


4.1 - Systems of Linear Equations in Two Variables
Determine whether (3, 9) is a solution of the following system.
5 x  2 y  3

 y  3x
5  3  2  9  3
9  3  3
15 18  3
3  3
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 x  y  8

x  3y  4
2  3   2  8
3  3 2  4
62 8
88
36  4
3  4
Both statements are not true, therefore (3, -2) is not a
solution to the given system of linear equations.
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by Graphing
 y3

 y  4 x  1

Solution :  1,3
33
3  4  1  1
33
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by Graphing
 x y 3

2 x  y  0
y  x 3

 y  2 x
Solution : 1, 2

1   2   3
2 1   2   0
33
00
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
7 5  39
8 2  33
10  6  36  20
27  6  50  29
15  3  6  12
17 12  14  9
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
 3x  y  1

4 x  y  6
3x  y  1
4x  y  6
7
7x
7x  7
x 1
3x  y  1
31  y  1
3 y 1
 y  2
y2
Solution
1,2
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
5 x  4 y  1
5 x  4 y  1
1

5   4 y  1
10
x

2
y

3

5
5 x  4 y  1
1  4 y  1
210 x  2 y  3
 4 y  2
5 x  4 y  1
20 x  4 y  6
25x  5
1
x
5
1
y
2
Solution
1 1
 , 
5 2
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
2 x  5 y  6

3 x  4 y  9
32 x  5 y  6
 23x  4 y  9
2x  5 y  6
2 x  50  6
2x  6
x3
6 x  15 y  18
Solution
 6 x  8 y  18
3,0
7y  0
y0
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
 4 x  7 y  10

 8 x  14 y  20
24 x  7 y  10
 8 x  14 y  20
8 x  14 y  20
 8 x  14 y  20
00
True Statement
Solution: All reals
Lines are the same
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by the Addition Method
(Also referred to as the Elimination Method)
3 x  9 y  5

 x  3y  2
False Statement
3x  9 y  5
 3x  3 y  2
lines are parallel
3x  9 y  5
 3 x  9 y  6
0  1
No Solution
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by Substitution
5 x  2 y  3

 y  3x
5 x  2 y  3
5x  23x   3
5x  6 x  3
 x  3
x3
y  3x
y  33
y9
Solution
3,9
4.1 - Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations by Substitution
2x  y  8
2x  y  8
2 x  y  8

2 3 y  4  y  8
2
x

0

8
x

3
y

4

 6y 8  y  8
2x  8
x  3y  4
7y 8  8
x4
7y  0
x  3 y  4
Solution
y0
4,0
4.1 - Systems of Linear Equations in Two Variables
Example
1
1
2
𝑥+ 𝑦=
2
3
3
LCD: 6
2
2
14
𝑥+ 𝑦=
3
5
15
LCD: 15
1
1
2
(6) 𝑥+(6) 𝑦= (6)
2
3
3
2
2
14
(15) 𝑥+(15) 𝑦= (15)
3
5
15
3𝑥 + 2𝑦=4
10𝑥 + 6𝑦=14
Solution
(2, −1)