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**5.1 Inverse & Joint Variation**

p.303 What is direct variation? What is inverse variation? What is joint variation?

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**Just a reminder… Direct Variation Use y=kx.**

Means “y varies directly with x.” k is called the constant of variation.

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**New stuff! Inverse Variation “y varies inversely with x.”**

k is the constant of variation.

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**Hint: Solve the equation for y and take notice of the relationship.**

Ex: tell whether x & y show direct variation, inverse variation, or neither. xy=4.8 y=x+4 Inverse Variation Hint: Solve the equation for y and take notice of the relationship. Neither Direct Variation

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**Ex: The variables x & y vary inversely, and y=8 when x=3.**

Write an equation that relates x & y. k=24 Find y when x= -4. y= -6

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**The inverse variation equation is y = When x = 2, y = 2 = 6. ANSWER**

The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x = 2. 4. x = 4, y = 3 y = a x Write general equation for inverse variation. 4 3 = a Substitute 3 for y and 4 for x. 12 = a Solve for a. 12 x The inverse variation equation is y = When x = 2, y = 2 = 6. ANSWER

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MP3 Players The number of songs that can be stored on an MP3 player varies inversely with the average size of a song. A certain MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB). Write a model that gives the number n of songs that will fit on the MP3 player as a function of the average song size s (in megabytes).

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**• Make a table showing the number of songs that will**

• Make a table showing the number of songs that will fit on the MP3 player if the average size of a song is 2MB, 2.5MB, 3MB, and 5MB as shown below. What happens to the number of songs as the average song size increases?

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**Write an inverse variation model. a n = s a 2500 = 4 10,000 = a**

STEP 1 Write an inverse variation model. a n = s Write general equation for inverse variation. a 2500 = 4 Substitute 2500 for n and 4 for s. 10,000 = a Solve for a. A model is n = s 10,000 ANSWER STEP 2 Make a table of values. From the table, you can see that the number of songs that will fit on the MP3 player decreases as the average song size increases. ANSWER

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The table compares the area A (in square millimeters) of a computer chip with the number c of chips that can be obtained from a silicon wafer. Computer Chips • Write a model that gives c as a function of A. • Predict the number of chips per wafer when the area of a chip is 81 square millimeters.

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SOLUTION STEP 1 Calculate the product A c for each data pair in the table. 58(448) = 25,984 62(424) = 26,288 66(392) = 25,872 70(376) = 26,320 Each product is approximately equal to 26,000. So, the data show inverse variation. A model relating A and c is: A c = 26,000 , or c = A 26,000 STEP 2 Make a prediction. The number of chips per wafer for a chip with an area of 81 square millimeters is 81 26,000 321 c =

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Joint Variation When a quantity varies directly as the product of 2 or more other quantities. For example: if z varies jointly with x & y, then z=kxy. Ex: if y varies inversely with the square of x, then y=k/x2. Ex: if z varies directly with y and inversely with x, then z=ky/x.

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**Examples: Write an equation.**

y varies directly with x and inversely with z2. y varies inversely with x3. y varies directly with x2 and inversely with z. z varies jointly with x2 and y. y varies inversely with x and z.

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**Write a general joint variation equation. z = axy**

The variable z varies jointly with x and y. Also, z = –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find z when x = 2 and y = 6. SOLUTION STEP 1 Write a general joint variation equation. z = axy Use the given values of z, x, and y to find the constant of variation a. STEP 2 –75 = a(3)(–5) Substitute 75 for z, 3 for x, and 25 for y. –75 = –15a Simplify. 5 = a Solve for a.

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STEP 3 Rewrite the joint variation equation with the value of a from Step 2. z = 5xy STEP 4 Calculate z when x = 2 and y = 6 using substitution. z = 5xy = 5(2)(6) = 60

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**Write an equation for the given relationship.**

y = a x a. y varies inversely with x. b. z varies jointly with x, y, and r. z = axyr c. y varies inversely with the square of x. y = a x2 d. z varies directly with y and inversely with x. z = ay x e. x varies jointly with t and r and inversely with s. x = atr s

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**Write a general joint variation equation. z = axy**

x = 4, y = –3, z =24 SOLUTION STEP 1 Write a general joint variation equation. z = axy Use the given values of z, x, and y to find the constant of variation a. STEP 2 24 = a(4)(– 3) Substitute 24 for z, 4 for x, and –3 for y. 24 = –12a Simplify. = a – 2 Solve for a.

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STEP 3 Rewrite the joint variation equation with the value of a from Step 2. z = – 2 xy STEP 4 Calculate z when x = – 2 and y = 5 using substitution. z = – 2 xy = – 2 (– 2)(5) = 20 z = – 2 xy ; 20 ANSWER

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**What is direct variation?**

y varies directly with x (y = kx) What is inverse variation? y varies inversely with x (y = k/x) What is joint variation? A quantity varies directly as the product of two or more other quantities ( y = kxy)

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Assignment p. 307 3-33 every third problem,

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