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Direct Variation and Proportion
Objective: Write and apply direct variation and proportion problems
Direct Variation
• The variable y varies directly as x if there is a nonzero constant k such that y = kx. The equation y = kx is called a direct variation equation and the number k is called the constant of variation.
Example 1
• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = -24 when x = 4.
Example 1
• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = -24 when x = 4.
• Use the direct variation equation• Substitute -24 for y and 4 for x 424
k
kxy
Example 1
• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = -24 when x = 4.
• Use the direct variation equation• Substitute -24 for y and 4 for x• Solve for k.
k
k
k
kxy
6
424
424
Example 1
• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = -24 when x = 4.
• Use the direct variation equation• Substitute -24 for y and 4 for x• Solve for k.
• The direct variation equation is
k
k
k
kxy
6
424
424
xy 6
Try This
• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = 21 when x = 7.
Try This
• Find the constant of variation, k and the direct variation equation if y varies directly as x and y = 21 when x = 7.
• Use the direct variation equation• Substitute 21 for y and 7 for x• Solve for k.
• The direct variation equation is
k
k
k
kxy
3
721
721
xy 3
Example 2
• Each day John rides his bike for exercise. When traveling at a constant rate, he rides 4 miles in about 20 minutes. At this rate, how long would it take John to travel 7 miles?
Example 2
• Each day John rides his bike for exercise. When traveling at a constant rate, he rides 4 miles in about 20 minutes. At this rate, how long would it take John to travel 7 miles?
Example 2
Proportions
• A ratio is a comparison of two numbers by division.• A proportion is a statement that two ratios are equal.
Proportions
• A ratio is a comparison of two numbers by division.• A proportion is a statement that two ratios are equal.
• A proportion of the form
can be rearranged like this:
d
c
b
a
bcad
bdd
c
b
abd
Cross-Product Property
• The result of what we just did is called the cross-product property of proportions. This is also called the means-extremes product property.
:0 and 0For db
bcadd
c
b
a then , If
Physics Application
Example 3
Example 3
Example 3
Example 4
Example 4
Class work
• In pairs/groups• Page 33• 8-13
Homework
• Pages 33-34• 15-25 odd• 37-51 odd