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Direct and Inverse Variation: Understanding the Relationship Between Variables

This article explores direct and inverse variation in mathematics, providing explanations, examples, and formulas to help understand the relationship between variables. Discover how to determine if variables vary directly or inversely, find the constant of variation, and solve for missing values. Learn about joint variation and how to write functions that model relationships between multiple variables.

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Direct and Inverse Variation: Understanding the Relationship Between Variables

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  1. 9-1/2-3 Notes Direct and Inverse Variation

  2. Direct Variation • Y varies directly to x, when x and y are related by the equation: y=kx. Here k is the constant of variation. • In other words…generally as x increases y will also increase and vice versa. Also, x and y are directly proportional so:

  3. Ex1: Determine from the table if y varies directly to x. If so, find the constant of variation. Ex2: D varies directly with P. Find the missing value. To solve for the missing value, set up a proportion…solve by cross multiplying. X and y are directly related because if you plug into the equation y=kx, you will get the same k value each time…k=1/5.

  4. Ex1: Determine from the table if y varies directly to x. If so, find the constant of variation. Ex2: D varies directly with P. Find the missing value. To solve for the missing value, set up a proportion…solve by cross multiplying. X and y are directly related because if you plug into the equation y=kx, you will get the same k value each time…k=1/5.

  5. Do numbers 1 to 4…Compare answers as a class.

  6. Key #1-4 • 1. Yes, x and y vary directly, because • 2. • 3. y = -6 • 4. y = 56/3

  7. Inverse Variation • Y varies inversely to x, when x and y are related by the equation • In other words…generally as x increases y will decrease and vise versa. Also x and y are inversely proportional so: (basically the x1 and y1 are diagonally across from each other, as are the x2 and y2) • This also means that

  8. Ex3: Determine from the table if y is inversely proportional to x. If so, find the constant of variation. Ex4: S varies inversely with T. Find the missing value. To solve for x, set up an inverse proportion and cross multiply. X and y are inversely related because when plugged into the equation y=k/x, you get the same thing every time for k. k=6.

  9. Ex3: Determine from the table if y is inversely proportional to x. If so, find the constant of variation. Ex4: S varies inversely with T. Find the missing value. To solve for x, set up an inverse proportion and cross multiply. X and y are inversely related because when plugged into the equation y=k/x, you get the same thing every time for k. k=6.

  10. Do numbers 5 to 8…Compare answers as a class.

  11. KEY #5-8 • 5. Yes, x and y are inversely proportional because • 6. y = 400 • 7. k = .6 (.4) = .24 • 8. x = 8

  12. Joint variation • Z varies jointly with x and y, when x, y , and z are related by the equation z=kxy. • ‘Varies’ tells you where to put the equal sign. k always comes after the equal sign.

  13. Example 5 • Z varies jointly as x and y, if z=56 when x=7 and y=10, find the constant of variation. • To solve use the joint variation equation z=kxy and solve for k.

  14. Example 6 Z varies directly with x and inversely with the cube of y. When x=8 and y=2, z=3. Find z when x=6 and y=4. • To solve you need to make an equation that relates x, y, and z. Remember the varies tells you where to put the equal sign, k always comes after the equal, directly means multiply, and inversely means divide. This means your equation should be: • Plug in the values they give for x, y, and z to solve for k. • Use this k value to solve for z when x=6 and y=4.

  15. Example 7 Describe the variation that is modeled by each formula. • Remember the equal sign is represented by varies when you are describing a variation! • If you are describing a variable that is multiplied you will say directly. • If you are describing a variable the is divided you will say inversely. • If you are describing two variables that are both multiplied say jointly. • If there is a number then it is the constant of variation! A varies jointly with b and h, when 0.5 is the constant of variation. V varies jointly with B and h, when 1/3 is the constant of variation.

  16. Example 8 z varies jointly with x and y and inversely with w. When x = 5, y = 6, and w = 2, z = 45. Write a function that models this relationship, then find z when x = 4, y = 8, and w = 16. • To solve you need to make an equation that relates x, y, and z. Remember the varies tells you where to put the equal sign, k always comes after the equal, directly means multiply, inversely means divide, jointly means multiply by both variables. This means your equation should be: • Then use the first set of values to solve for the k value: • Then plug in the second set of values with k to solve for z:

  17. Do numbers 9 to 20…Compare as a class.

  18. Answers. • 1. yes; k=5 • 2. k=27/19 • 3. y=-6 • 4. y=56/3 • 5. yes; k=24 • 6. y=400 • 7. k=6/25 • 8. x=8 • 9. k=-1 • 10. z=(0.5y)/x • 11. directly; k=5 • 12. b • 13. y=-22 • 14. x=100/7 • 15. k=3 • 16. 5 hours • 17. 14 days • 18. l varies directly with V and inversely with the product of w and h. • 19. z=4/3 • 20. k=4186

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