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Understanding Direct and Inverse Variation with Examples

This guide explains direct and inverse variation with practical examples. It describes how to identify the type of variation between two variables, x and y, based on given equations. You'll learn to write inverse variation equations, analyze how changes in one variable affect the other, and solve for unknowns. Using an MP3 player as a context, the guide illustrates how the number of songs that can be stored varies inversely with song size. Through examples and practice problems, you'll solidify your understanding of this important mathematical concept.

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Understanding Direct and Inverse Variation with Examples

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  1. y = y c. = x 4 7 x EXAMPLE 1 Classify direct and inverse variation Tell whether xand yshow direct variation, inverse variation, or neither. Type of Variation Given Equation Rewritten Equation a.xy = 7 Inverse b.y = x + 3 Neither Direct y = 4x

  2. y= 7= ANSWER 28 The inverse variation equation is y = x 28 = –14. Whenx = –2, y = a a –2 4 x EXAMPLE 2 Write an inverse variation equation The variables xand yvary inversely, and y = 7 when x=4. Write an equation that relates xand y. Then find ywhen x = –2 . Write general equation for inverse variation. Substitute 7 for yand 4 for x. 28 = a Solve for a.

  3. MP3Players 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). EXAMPLE 3 Write an inverse variation model • Write a model that gives the number nof songs that will fit on the MP3 player as a function of the average song size s(in megabytes).

  4. • 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? EXAMPLE 3 Write an inverse variation model

  5. STEP 1 Write an inverse variation model. a n= s a 2500= 4 ANSWER 10,000 s A model is n = EXAMPLE 3 Write an inverse variation model Write general equation for inverse variation. Substitute 2500 for n and 4 for s. 10,000 = a Solve for a.

  6. STEP 2 Make a table of values. ANSWER 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. EXAMPLE 3 Write an inverse variation model

  7. 0.75 y = x for Examples 1, 2 and 3 GUIDED PRACTICE Tell whether xand yshow direct variation, inverse variation, or neither. Type of Variation Given Equation Rewritten Equation Direct 1. 3x = y y = 3x 2.xy = 0.75 Inverse Neither 3.y = x –5

  8. a y= x a 3= 4 ANSWER 12 The inverse variation equation is y = x 12 = 6. Whenx = 2,y = 2 for Examples 1, 2 and 3 GUIDED PRACTICE The variables xand yvary inversely. Use the given values to write an equation relating xand y. Then find ywhen x=2. 4.x = 4,y = 3 Write general equation for inverse variation. Substitute 3 for yand 4 for x. 12 = a Solve for a.

  9. a y= x a –1= 8 ANSWER – 8 The inverse variation equation is y = x – 8 = – 4. Whenx = 2,y = 2 for Examples 1, 2 and 3 GUIDED PRACTICE 5.x = 8,y = –1 Write general equation for inverse variation. Substitute –1 for yand 4 for x. – 8 = a Solve for a.

  10. , 6.x = y = 12 a y= x 1 a Substitute 12 for yand for x. 12= 1 2 2 ANSWER 6 The inverse variation equation is y = x 6 = 3. Whenx = 2,y = 2 1 2 for Examples 1, 2 and 3 GUIDED PRACTICE Write general equation for inverse variation. 6 = a Solve for a.

  11. 7. What If? In Example 3, what is a model for the MP3 player if it stores 3000 songs when the average song size is 5MB? Write an inverse variation model. a n= s a 3000= 5 15,000 s ANSWER A model is n = for Examples 1, 2 and 3 GUIDED PRACTICE Write general equation for inverse variation. Substitute 3000 for n and 5 for s. 15,000 = a Solve for a.

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