Direct Variation

Direct Variation A function of the form y = kx, k ≠ 0 and k is the constant of variation. The variables x and y are said to vary directly with each other.

Identifying a direct variation given an equation. Example: Is each equation a direct variation? If yes, find the constant of variation. 5x + 2y = 0 Solution: 2y = -5x subtract 5x from both sides y = (-5/2)x divide both sides by 2. Since the equation is in the form y = kx, the equation is a direct variation. The constant of variation k is -5/2.

Identifying a direct variation given an equation. 5x + 2y = 9 Solution: 2y = -5x + 9 subtract 5x from both sides y = -5/2x +9/2 divide both sides by 2 Since the equation is not in the form y = kx, then it is not a direct variation. Clue: If b is not equal to zero, it is not a direct variation.

Do these… Is each equation a direct variation? If it is, find the constant of variation. 7y = 2x 3y + 4x = 8 Y-7.5x = 0 Answers: 1. yes; 2/7 2. No 3. Yes; 7.5

Writing an Equation of a Direct Variation given a Point Example: Write an equation of a direct variation that includes the point (4, -3). Solution: y = kx Start with the function form of a direct variation -3=k(4) substitute 4 for x and -3 for y -3/4 = k divide both sides by 4 to solve for k y = -3/4x write an equation. Substitute -3/4 for k in y = kx

Do these… Write an equation of the direct variation that includes the following points: (-3, -6) (12, -4) (-1/2, 5/3) Answer: 1. y = 2x 2. y = -1/3x 3. y = -10/3x

Table of Values for a Direct Variation Example: For each table, use the ratio y/x to tell whether y varies direct with x. If it does, write an equation for the direct variation. 1. Check for y/x if they are equal. x y -3 2.25 1 -0.75 4 6 -4.5 Since the ratio y/x are equal for each ordered pair, it is a direct variation and the constant of variation is -3/4 or 0.75. The equation is y = -3/4x

Table of Values for a Direct Variation 2. Solution: x y 2 -1 4 1 6 3 9 4.5 No, it is not a direct variation since y/x is not the same for all pairs of data.

Do these… For the data in each table, tell whether y varies directly with x. If it does, write an equation for the direct variation. 1. 2. 3. x y 4 8 7 14 10 20 x y -3 -2 3 2 9 6 x y 4 3 5 4.5 11 13.5 Answers: 1. yes, y = 2x 2. yes, y = 2/3x 3. no

Finding an Unknown Value in a Direct Variation The ordered pairs (3, 2)and (6, y) are for the same direct variation. Find the missing value. Solution: There are two ways of solving for the unknown Method 1 Using Proportion (3, 2) and (6, y) since a direct variation has constant ratio then 12 = 3y cross multiply 4 = y divide both sides by 3 to solve for y

Finding an Unknown Value in a Direct Variation Solution: Method 2 Use the formula for direct variation y = kx, then solve for k (3, 2) and (6, y) y = kx 2 = k(3) 2/3 =k Y = 2/3x Solve for y using y = 2/3x and (6, y) y =2/3x y =(2/3)(6) substitute x by 6 y = 4

Do these… The ordered pairs in each exercise are for the same direct variation. Find each missing value. Use any method. (-2, 8) and (x, 12) (4, y) and (16, 12) (4.8, 5) and (2.4, y) Answers: 1. x = -3 2. y = 3 3. y = 2.5

Real-World Problem Solving The force you must apply to lift an object varies directly with the object’s weight. You would need to apply 0.625 lb of force to a windlass to lift a 28-lb weight. How much force would you need to lift 100 lb? Solution: Relate: A force of 0.625 lb lifts 28 lbs. What force lifts 100 lb? Define: Let n = the force you need to lift 100 lb Write: use proportion

Cont… Write: use proportion substitute using the given values cross multiply divide both sides by 28 to solve for n You need about 2.2 lb of force to lift 100 lb.

Do these… 1. Charles’ Law states that at constant pressure, the volume of a fixed amount of gas varies directly with its temperature measured in degrees Kelvin. A gas has a volume of 250 mL at 300K. a. Write an equation for the relationship between volume and temperature. b. What is the volume if the temperature increases to 420K. Answers: 1 a. Let y = volume x = temperature; y = 5/6x b. 350mL

Do these… 2. Your percent grade varies directly with the number of correct answers. You got a grade of 80 when you had 20 correct answers. a. Write an equation for the relationship between percent grade and number of correct answers. b. What would your percent grade be with 24 correct answers. 3. The amount of simple interest earned in a savings account varies directly with the amount of money in the savings account. You have $1000 in you savings account and earn $50 simple interest. How much interest you earn if you had $1500 in your savings account? Answers: 2.a. Y = 4x; y = grade x = correct ans. b. 96 3. $75