Direct
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Transcript of Direct
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The general equation for DIRECT VARIATION is y = kx with k 0.
k is called the constant of variation.
We will do an example together.
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If y varies directly as x, and y=24 and x=3 find: (a) the constant of variation
(b) Find y when x=2
(a) Find the constant of variation
y kx Write the general equation
24 k 3 Substitute
k 8
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(b) Find y when x=2
First we find the constant of variation, which was k=8
Now we substitute into y=kx.
y kx
y 82y 16
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Another method of solving direct variation problems is to use proportions.
If y1 = kx1, then k =y1
x1
and
If y 2 kx2 , then k =y2
x2
Therefore...
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y1
x1
y2
x2
So lets look at a problem that can by solved by either of these two methods.
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If y varies directly as x and y=6 when x=5, then find y when x=15.
Proportion Method:6
5
y
15Let x1 5, y1 6, x2 15, y2 y
5y 90
y 18
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Now lets solve using the equation.
y kx
6 k 5
k 6
5
y kx
y 6
515
y 18
Either method gives the correct answer, choose the easiest for you.
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Now you do one on your own.
y varies directly as x, and x=8 when y=9. Find y when x=12.
Answer: 13.5
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What does the graph y=kx look like?A straight line with a y-intercept of 0.
5
-5
-10 10
f x = 3 x
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Inverse Variation
y varies inversely as x if k 0
such that xy=k or y k
x
Just as with direct variation, a proportion can be set up solve problems of indirect variation.
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x1
y2
x2
y1
A general form of the proportion
Lets do an example that can be solved by using the equation and the proportion.
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Find y when x=15, if y varies inversely as x and x=10 when y=12
Solve by equation:
xy k
10 12 k
120 k
xy k
15 y 120y 8
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Solve by proportion:
x1
y2
x2
y1
15
12
10
y
15y 120
y 8
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Solve this problem using either method.
Find x when y=27, if y varies inversely as x and x=9 when y=45.
Answer: 15
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Joint Variation
For three quantities x, y and z, if there is a constant k such that
z = kxy
We say “z varies jointly as y and x” or
“z is jointly proportional to x and y”.
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A general form of the proportion
Lets do an example that can be solved by using the equation and the proportion.
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Example 9
• U varies jointly as V and the square of W. if V =4 and W = 3, then U = 18. Find the value of V when U = 24 and W = 3.
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Class work…
• Work book page 123 and 124 even