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Direct Variation Algebra I Mastery Cate Condon The store you go into in the mall sells t-shirts. You are looking around and you see that 3 t-shirts cost $15 total. Given this information how much would 5 t-shirts cost? O We can set up a proportion: 3 5 = 15 c O Cross multiply: 3c = 5*15= 75 3c = 75 c = 25 O 3c = 75 O We know that c is the variable. What does the c stand for or represent? O What is the 3 in front of the c called? O 3c = 75 O We know that 3 is called the coefficient of the variable. O Another name for this is the constant of variation. O Why do you think that a number in front of variable is called a constant of variation? O 10 = 2x 2x = 10 O For this equation, what is the variable and what is the constant of variation? O 3x = 1/6 O For this equation, what is the variable and what is the constant of variation? O 10 = 2x O We can write this equation substituting variables: y = kx where k is the constant of variation and x and y are variables O We call this equation, y = kx, direct variation Direct Variation O A direct variation is a function in the form y = kx where k does not equal 0. O An equation is a direct variation if the equation can be written in the form y = kx. Is the Equation a Direct Variation? If it is, find the constant of variation. O -6x + 2y = 0 Want to see if the equation can be written like y=kx O Solve for y: 2y = 6x y = 3x O Yes it is a direct variation. O The constant of variation is 3 Is the Equation a Direct Variation? If it is, find the constant of variation O 7y = 2x O Solve for y: y = (2/7) x O Yes it is a direct variation. O The constant of variation is (2/7) Try These O 0 = 10 + 3x O 8x = 4y O 6 + 9x = 2y Direct Variation O We say that y varies directly with x when we have the equation y = kx. O So if I say that p varies directly with z, what would our equation be? (keep k as our constant) Real World Example O Your distance from lightning varies directly with the time it takes you to hear thunder. If you hear thunder 10 seconds after you see the lightning, you are about 2 miles from the lightning. Write an equation for the relationship between time and distance. Real World Example O Your distance from lightning varies directly with the time it takes you to hear thunder: O y = distance from lightening O x = time it takes you to hear thunder Real World Example O y = kx O We know that y is 2 miles from the lightning (y = distance from lightening O We know that x is 10 seconds (x = time it takes you to hear thunder) O 2 = k(10) solve for k O k = (2/10)=(1/5) O Our equation: y = (1/5)x Real World Example y x k Case 1 2 10 1/5 Case 2 3 15 1/5 Case 3 4 20 1/5 Case 4 5 25 1/5 DV equation y= (1/5)*10 y= (1/5)*15 y= (1/5)*20 y= (1/5)*25 Real World Example 30 25 20 y x k 15 10 5 0 0 1 2 3 4 5 Real World Example Real World Example Try This O A recipe for a dozen corn muffins calls for 1 cup of flower. The number of muffins varies directly with the amount of flour you use. Write a direct variation for the relationship between the number of cups of flour and the number of muffins Real World Example O y = kx O y = the number of muffins O x = amount of flour used O 12 = k(1) 12 = k O So our equation is y = 12x Closing O p varies directly with z. If p = 210 when z = 200, then write the formula for the relation between p and z. O Work by yourself on the notecard on your desk for the last 5 minutes.