1. Inverse Variation
ALGEBRA 1 LESSON 8-10
(For help, go to Lesson 5-5.)
Suppose y varies directly with x. Find each constant of variation.
1. y = 5x 2. y = –7x 3. 3y = x y = x
Write an equation of the direct variation that includes the given point.
5. (2, 4) 6. (3, 1.5) 7. (–4, 1) 8. (–5, –2)
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2. Inverse Variation Solutions 1. y = 5x; constant of variation = 5
ALGEBRA 1 LESSON 8-10
1. y = 5x; constant of variation = 5
2. y = –7x; constant of variation = –7
y = x
• 3y = • x
y = x; constant of variation =
y = x
y = x
4 • y = 4 • x
y = 4x; constant of variation = 4
Solutions 1 3 1 3 1 3 1 3 1 4 1 4
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3. Inverse Variation Solutions (continued)
ALGEBRA 1 LESSON 8-10
Solutions (continued)
5. Point (2, 4) in y = kx: 4 = k(2), so k = 2 and y = 2x.
6. Point (3, 1.5) in y = kx: 1.5 = k(3), so k = 0.5 and y = 0.5x.
7. Point (–4, 1) in y = kx: 1 = k(–4), so k = – and y = – x.
8. Point (–5, –2) in y = kx: –2 = k(–5), so k = and y = x. 1 4 1 4 2 5 2 5
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4. Inverse Variation
ALGEBRA 1 LESSON 8-10
Suppose y varies inversely with x, and y = 9 when x = 8. Write an equation for the inverse variation.
xy = k Use the general form for an inverse variation.
(8)(9) = k Substitute 8 for x and 9 for y.
72 = k Multiply to solve for k.
xy = 72 Write an equation. Substitute 72 for k in xy = k.
The equation of the inverse variation is xy = 72 or y = . 72 x
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5. Inverse Variation
ALGEBRA 1 LESSON 8-10
The points (5, 6) and (3, y) are two points on the graph of an inverse variation. Find the missing value.
x1 • y1 = x2 • y2 Use the equation x1 • y1 = x2 • y2 since you know coordinates, but not the constant of variation.
5(6) = 3y2 Substitute 5 for x1, 6 for y1, and 3 for x2.
30 = 3y2 Simplify.
10 = y2 Solve for y2.
The missing value is 10. The point (3, 10) is on the graph of the inverse variation that includes the point (5, 6).
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6. Inverse Variation
ALGEBRA 1 LESSON 8-10
Jeff weighs 130 pounds and is 5 ft from the lever’s fulcrum. If Tracy weighs 93 pounds, how far from the fulcrum should she sit in order to balance the lever?
Relate:  A weight of 130 lb is 5 ft from the fulcrum. A weight of 93 lb is x ft from the fulcrum. Weight and distance vary inversely.
Define:  Let weight1 = 130 lb Let weight2 = 93 lb Let distance1 = 5 ft Let distance2 = x ft
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7. Inverse Variation Write: weight1 • distance1 = weight2 • distance2
ALGEBRA 1 LESSON 8-10
(continued)
Write: weight1 • distance1 = weight2 • distance2
130 • 5 = 93 • x Substitute.
650 = 93x Simplify.
6.99 = x Simplify.
= x Solve for x.
650 93
Tracy should sit 6.99, or 7 ft, from the fulcrum to balance the lever.
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8. Inverse Variation
ALGEBRA 1 LESSON 8-10
Decide if each data set represents a direct variation or an inverse variation. Then write an equation to model the data.
The values of y seem to vary inversely with the values of x.
x y
3 10
5 6
10 3 a.
Check each product xy.
xy: 3(10) = 30    5(6) = 30    10(3) = 30
The product of xy is the same for all pairs of data.
So, this is an inverse variation, and k = 30.
The equation is xy = 30.
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9. Inverse Variation
ALGEBRA 1 LESSON 8-10
(continued)
x y
2 3
4 6
8 12 b.
The values of y seem to vary directly with the values of x.
Check each ratio . y x 6 4
= 1.5 12 8 y x 3 2
The ratio is the same for all pairs of data. y x
So, this is a direct variation, and k = 1.5.
The equation is y = 1.5x.
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10. Inverse Variation
ALGEBRA 1 LESSON 8-10
Explain whether each situation represents a direct variation or an inverse variation.
a. You buy several souvenirs for $10 each.
The cost per souvenir times the number of souvenirs equals the total cost of the souvenirs.
Since the ratio is constant at $10 each, this is a direct variation.
cost
souvenirs
b. The cost of a $25 birthday present is split among several friends.
The cost per person times the number of people equals the total cost of the gift.
Since the total cost is a constant product of $25, this is an inverse variation.
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11. Inverse Variation
ALGEBRA 1 LESSON 8-10
1. The points (5, 1) and (10, y) are on the graph of an inverse variation. Find y.
2. Find the constant of variation k for the inverse variation where a = 2.5 when b = 7.
0.5
17.5
x y 1 2 6 3 9
3. Write an equation to model the data and complete the table.
xy = 1 3 18
4. Tell whether each situation represents a direct variation or an inverse variation.
a. You buy several notebooks for $3 each.
b. The $45 cost of a dinner at a restaurant is split among several people.
direct variation
Inverse variation
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