1. PRE-ALGEBRA

2. Lesson 8-4 Warm-Up

3. PRE-ALGEBRA What is a “direct variation”? How do you find the constant, k, of a direct variation given a point on its line? direct variation (sometimes called a direct proportion) - a linear (line) function (relationship in which y varies, or “changes”, directly with changes in x) that follows the equation: y = kx where the coefficient k is called the “constant of the variation” (a constant is a number that never changes) and k ≠ 0. Note: Since y = 0 when x = 0, all direct variations pass through the origin (0, 0) Examples: y = ¾ x y = -½ x To find the k of a a direct variation, substitute the (x, y) values of a point on its line. Example: Find the constant of variation, k, of the direct variation that contains the point (2, 4) y = kxDirect Variation Equaton. 4 = k(2)Substitute the x and y values of (2,4) into y = kx. 4 = 2kSimplify. 2 2Solve for k by dividing both sides by 2. 2 = k The constant of variation is 2. Direct Variation (8-4)

4. PRE-ALGEBRA The table of values models a direct variation. Graph the direct variation. Find the constant of variation. x –2 0 2 6 y –1 0 1 3 Plot the points on a coordinate plane. Connect the points. Direct Variation LESSON 8-4 Additional Examples

5. PRE-ALGEBRA (continued) y = kx Use the equation for a direct variation. 1 = k 2 Substitute a point in the table like (2, 1). The constant of variation is. 1212 = k Solve for k. 1212 Use a point to find the constant of variation. Direct Variation LESSON 8-4 Additional Examples

6. PRE-ALGEBRA How can use a direct variation to solve an equation in which there is a proportional relationship between two things? You can use a direct proportion equation to solve any problem in which there is a proportional relationship between two values (x and y). To do this, find the constant of variation using the given relationship between x and y. Then, you can use substitution to find how the the change in on value affects the other value. Example: Your weight on Earth’s surface varies directly with your weight on Mars. If a person weighing 120 lb. on Earth weighs about 40 lb. on Mars, how much would a person weighing 180 lb. on Earth weigh on Mars? Step 1: Find the constant of variation. Let E = weight on Earth Assign variables to weight on Earth and Let M = weight on Mars Mars E = kM Write the equation for a direct variation using your variables. 120 = k 40Substitute 120 for E and 40 for M 40 40Divide each side by 40 to isolate the k. 3 = kSimplify. Direct Variation (8-4)

7. PRE-ALGEBRA Step 2: Find the weight of a 180 pound person on Mars. E= kM Write the equation for a direct variation using your variables. 180 = 3 · MSubstitute 180 for E and 3 for k 3 3Divide each side by 3 to isolate the M 60 = MSimplify. A person weighing 180 pounds on Earth weights about 60 pounds on Mars. Direct Variation (8-4)

8. PRE-ALGEBRA The number of miles Joe walks varies directly with the number of hours he walks. He walks 6 miles in 2 hours. Write a direct variation to find how many hours it takes Joe to walk 15 miles. Step 1Find the constant of variation. m = kh Write the equation for a direct variation. 6 = k 2 Substitute 6 for m and 2 for h. 3 = k Divide each side by 2. Simplify. Direct Variation LESSON 8-4 Additional Examples

9. PRE-ALGEBRA Step 2Write the equation using the value of k. m = kh Write the equation for a direct variation. 15 = 3h Substitute 15 for m and 3 for k. 5 = h Divide each side by 3. Simplify. It takes Joe 5 hours to walk 15 miles. (continued) Direct Variation LESSON 8-4 Additional Examples

10. PRE-ALGEBRA Step 1Find the constant of variation. y = kx Write the equation for a direct variation. – 5 = k(7) Substitute –5 for y and 7 for x. Step 2Write the equation using the value of k. y = kx Write the equation for a direct variation. k = – Divide each side by 7. Simplify. 5757 y = – x Replace k with –. 5757 5757 Write an equation for the direct variation that includes B (7, –5). Direct Variation LESSON 8-4 Additional Examples

11. PRE-ALGEBRA How do you tell whether each “data pair” (x and y) in a table is a direct variation? Tip: You can write y = kx as k = y / x if you divide both sides by x. If each data pair (x  y) equals k (in other words, the ratio of y to x is the same for each x and y pair), then the table represents a direct variation. Example: Is the following table a direct variation? No, the ratio of (in other words, the “k”) is not the same for all of the x and y data pairs. Direct Variation (8-4)

12. PRE-ALGEBRA 1 –2 = –0.5 –1 2 –2 4 = –0.5 yxyx xy –2 1 2–1 4–2 2 –1 = –2 2121 –4 2 = 2 = –2 yxyx xy –1 2 12 2–4 For the data in each table, use the ratio to tell whether y varies directly with x. If it does, write an equation for the direct variation. Yes, the constant of variation is –0.5. The equation is y = –0.5x. yxyx No, the ratio is not the same for each pair of data. yxyx a.b. Direct Variation LESSON 8-4 Additional Examples Direct Variation LESSON 8-4

13. PRE-ALGEBRA Write an equation for a direct variation that includes each point. 1.R(3,5) 2.S(6, – 2)3.T(– 1, –2) 4. Graph the direct variation modeled by the table of values. Find the constant of variation. 5. The number of words Mary types varies directly with the number of minutes she types. She types 90 words in 3 minutes. Write a direct variation to find how many minutes it takes Mary to type 60 words. y = 2x w = 30m; 2 minutes y = x 5353 y = – x 1313 Direct Variation LESSON 8-4 Lesson Quiz