Direct Variation and Proportion Objective: Write and apply direct variation and proportion problems.
Direct Variation Learn to recognize direct variation and identify the constant of proportionality.
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Transcript of Direct Variation Learn to recognize direct variation and identify the constant of proportionality.
Direct Variation
Learn to recognize direct variation and identify the constant of proportionality.
Spiders
How many legs does a spider have? 8 legs Therefore
2 spiders have a total of 16 legs3 spiders have a total of 24 legs4 spiders have a total of 32 legsAnd so on. . .
This type of relationship is called . . .
Direct Variation
The relationship between the amount of spiders and how many legs they have can be said to vary directly!
We will be learning about equations, tables, and graphs of direct variations.
Sometimes this is called direct proportion rather than direct variation but it is the same thing.
Direct Variation
A direct variation relationship can be represented by a linear equation in the form y = kx, where k is a positive number called the constant of proportionality.
The constant of proportionality can sometimes be referred to as the constant of variation.
y = kx When two variable quantities have a constant
(unchanged) ratio, their relationship is called a direct proportion.
We say, “y varies directly as x.” k is the constant of proportionality which
means it never changes within a problem.
Finding Values Using a Direct Variation Equation
At a frog jumping contest, Edward’s frog jumped 60 inches. Bella’s frog jumped 72 inches. Jacob’s frog jumped 6.5 feet. Use the equation y = 12x, where y is the number of inches and x is the number of feet to find the missing values of the table.
Edward’s Frog Bella’s Frog Jacob’s Frog
Inches (y) 60 72 ___
Feet (x) ___ 6 6.5
Using Equations and Tables
Edward’s Frog Bella’s Frog Jacob’s Frog
Inches (y) 60 72 ___
Feet (x) ___ 6 6.5
xy 12 xy 12
x1260 5.612ySubstitute
12
12
12
60 x 78ySolve
x5 Simplify
78
5
Now You Try
Identify the constant of proportionality:
1. y = 15x
15
2. y = .72x
.72
3. y = ¼x
¼
How about solving for k?
When we are given the x and y values, we can solve for the constant of proportionality.
Example If y varies directly with x, and y = 8 when x = 12,
find k and write an equation that expresses this variation.
Steps: 8 = k × 12 Substitute numbers into y = kx 8/12 = (k × 12)/12 Divide both sides by 12 2/3 = k Simplify y = 2/3x Plug k back into the equation
Now You Try
If y varies directly as x, and x = 12 when y = 9, what is the equation that describes this direct variation?
y = kx 9 = k × 12 9/12 = k ¾ = k y = ¾x
Now You Try
If y varies directly as x, and x = 5 when y = 10, what is the equation that describes this direct variation?
y = kx 10 = k × 5 10/5 = k 2 = k y = 2x
Copy and fill out tables:
x y
1
2
3
4
5
x y
1
2
3
4
5
y = x y = 4x
1
2
3
4
5
4
8
12
16
20
Copy and fill out tables:
x y
1
2
3
4
5
x y
1
2
3
4
5
y = 10x y = ½x
10
20
30
40
50
½
1
3/2
2
5/2
Direct Variation Equation
Sometimes we will have to put an equation into y = kx form and solve for y.
Then we will be asked to identify k, the constant of proportionality.
Examples: Tell whether each
equation or relationship is a direct variation. If so identify the constant of proportionality.
1. 4y = 2x
2. ½y – ¾x = 0
3. 7y – 5 = 3x
More Examples
If y varies directly as x and y = 24 when x = 16, find y when x = 12.
Solution: Set up a proportion since the ratios of corresponding values of x to y are always the same.
1216
24 y
y 161224y16288
16
16
16
288 y
y18