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Mastering Direct, Inverse, and Joint Variations in Algebra

Learn to simplify and solve problems involving direct, inverse, and joint variations in algebra using constant of variation. Explore examples and practice finding k values in practical scenarios.

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Mastering Direct, Inverse, and Joint Variations in Algebra

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  1. 9.5 OBJECTIVE PAP Algebra 2 NOTES 9.4 TLW… Simplify and work problems dealing with direct, inverse, and joint variations.

  2. K = Constant of Variation Direct Variation The equation is: How to find k: Inverse Variation The equation is: How to find k: Y=kx

  3. K = Constant of Variation Joint Variation Joint variation is the same as direct variation with two or more quantities. • Examples of Joint Variation • y = 7xz, here y • y = 7x2z3, here y • Area of a triangle =           • Here, A varies jointly with x and z (k=7) varies jointly with x2 and z3 (k=7). Varies jointly with base ‘b’ and height ‘h’ (k=1/2)

  4. Given: y varies directlywith x and y = 12 when x = -4 • What is the constant • of variation (k) ? 2. Find y when x = 5

  5. Find k in each Inverse Variation

  6. (5,10) and (2,y) are from the same inverse variation. Find y.

  7. x and y vary inversely where x=-2 & y=1/2 Write a function to model inverse variation. Find y if x=1/2

  8. If z varies directly with x and inversely with y, the equation for this is: x y

  9. Suppose that z varies jointly with x and y when x = 2 and y = 3, then z = 60. A. Write the function that models the variation. Find the value of “k” first Now you can write the equation B. Find the value of z when x = 4 and y = -10

  10. Is the relationship between the variables direct or inverse? Find “k” using both equations. If the value of “k” remains the same each time, then it represent that variation. Direct : y = kx so k = y/x Inverse: y = k/x so k = xy You Can Stop Since #’s are not the same!! You Can Stop Since #’s are not the same!! Neither

  11. Is the relationship between the variables direct or inverse? Inverse: y = k/x so k = xy Direct : y = kx so k = y/x Direct Please remember that the order pair (0,0 ) is a direct variation point!!! Find “k” using both equations. If the value of “k” remains the same each time, then it represent that variation.

  12. Describe the combined variation that is modeled by the formula: Molar Mass • Molar mass varies jointly/directly with density and temperature and inversely with pressure BTW… k = 62.4

  13. Describe the combined variation that is modeled by the equation: • y varies jointly/directly with the square of w and square root of x and inversely with cube of z BTW… k = 2/3

  14. If m varies directly as the square root of y, inversely as p5, and directly as n, what happens to m when y is multiplied by 9, p is multiplied by 2 and n is quadrupled?: BTW… k = 1 m is multiplied by Equation:

  15. A drama club is planning a bus trip to NYC to see a Broadway play. The cost per person for the bus rental varies inversely as the number of people going on the trip. It will cost $50 per person if 20 people go on the trip. To the nearest dollar, how much will it cost per person if 100 people go on the trip?

  16. Suppose that m and r vary inversely and that r=4/9 when m=6. Write a function that models the inverse variation and find r when m=2

  17. The amount of oil used by a ship travelling at a uniform speed varies jointly with the distance and the square of the speed. The ship uses 500 barrels of oil in travelling 50mi at 30mph. Write an equation that models this variation and find k. Also find how many barrels of oil are used when the ship travels 10mi at 45mph.

  18. Homework #5 Due next class Your next quiz will cover Problems from Notes 9.3 Simplifying Rational Expressions

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