Pre-Algebra 11-5 Direct Variation Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

Pre-Algebra 11-5 Direct Variation

Direct Variation 11-5 Helpful Hint Pre-Algebra 11-5 Direct Variation The graph of a direct-variation equation is always linear and always contains the point (0, 0). The variables x and y either increase together or decrease together. Helpful Hint

Example 1B: Determining Whether a Data Set Varies Directly Pre-Algebra 11-5 Direct Variation Example 1B: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. B.

Direct Variation 11-5 Example 1B Continued Pre-Algebra 11-5 Direct Variation Example 1B Continued Make a graph that shows the relationship between the number of minutes and the distance the train travels. Plot the points. The points lie in a straight line. (0, 0) is included.

Direct Variation 11-5 Example 1B Continued Pre-Algebra 11-5 Direct Variation Example 1B Continued You can also compare ratios to see if a direct variation occurs. 25 10 50 20 75 30 100 40 Compare ratios. = = = The ratios are proportional. The relationship is a direct variation.

Example 1A: Determining Whether a Data Set Varies Directly Pre-Algebra 11-5 Direct Variation Example 1A: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. A.

Direct Variation 11-5 Example 1A Continued Pre-Algebra 11-5 Direct Variation Example 1A Continued Make a graph that shows the relationship between Adam’s age and his length.

Direct Variation 11-5 Example 1A Continued Pre-Algebra 11-5 Direct Variation Example 1A Continued You can also compare ratios to see if a direct variation occurs. 81 81 ≠ 264 The ratios are not proportional. 22 3 27 12 = ? 264 The relationship of the data is not a direct variation.

Example 2A: Finding Equations of Direct Variation Pre-Algebra 11-5 Direct Variation Example 2A: Finding Equations of Direct Variation Find each equation of direct variation, given that y varies directly with x. A. y is 54 when x is 6 y = kx y varies directly with x. 54 = k  6 Substitute for x and y. 9 = k Solve for k. Substitute 9 for k in the original equation. y = 9x

Example 2B: Finding Equations of Direct Variation Pre-Algebra 11-5 Direct Variation Example 2B: Finding Equations of Direct Variation B. x is 12 when y is 15 y = kx y varies directly with x. 15 = k  12 Substitute for x and y. = k 5 4 Solve for k. Substitute for k in the original equation. 5 4 y = x 5 4

Example 2C: Finding Equations of Direct Variation Pre-Algebra 11-5 Direct Variation Example 2C: Finding Equations of Direct Variation C. y is 8 when x is 5 y = kx y varies directly with x. 8 = k  5 Substitute for x and y. = k 8 5 Solve for k. Substitute for k in the original equation. 8 5 y = x 8 5

Pre-Algebra 11-5 Direct Variation

Direct Variation 11-5 Try This: Example 1B Pre-Algebra 11-5 Direct Variation Try This: Example 1B Determine whether the data set shows direct variation. B. Ounces in a Cup Ounces (oz) 8 16 24 32 Cup (c) 1 2 3 4

Try This: Example 1B Continued Pre-Algebra 11-5 Direct Variation Try This: Example 1B Continued Make a graph that shows the relationship between ounces and cups. Number of Cups Number of Ounces 2 3 4 8 16 24 1 32 Plot the points. The points lie in a straight line. (0, 0) is included.

Try This: Example 1B Continued Pre-Algebra 11-5 Direct Variation Try This: Example 1B Continued You can also compare ratios to see if a direct variation occurs. = 1 8 2 16 3 24 4 32 Compare ratios. The ratios are proportional. The relationship is a direct variation.

Direct Variation 11-5 Try This: Example 1A Pre-Algebra 11-5 Direct Variation Try This: Example 1A Determine whether the data set shows direct variation. A. Dan's Basketball Shots  Distance (ft) 20 30 40 Number of Baskets 5 3

Try This: Example 1A Continued Pre-Algebra 11-5 Direct Variation Try This: Example 1A Continued Make a graph that shows the relationship between number of baskets and distance. 5 4 3 Number of Baskets 2 1 20 30 40 Distance (ft)

Direct Variation 11-5 Try This: Example 1A Pre-Algebra 11-5 Direct Variation Try This: Example 1A You can also compare ratios to see if a direct variation occurs. 60 150 ≠ 60. The ratios are not proportional. 5 20 3 30 = ? 150 The relationship of the data is not a direct variation.

Direct Variation 11-5 Try This: Example 2A Pre-Algebra 11-5 Direct Variation Try This: Example 2A Find each equation of direct variation, given that y varies directly with x. A. y is 24 when x is 4 y = kx y varies directly with x. 24 = k  4 Substitute for x and y. 6 = k Solve for k. Substitute 6 for k in the original equation. y = 6x

Example 3: Money Application Pre-Algebra 11-5 Direct Variation Example 3: Money Application Ms. Cason has $4000 in a CD and $4000 in a money market account. The amount of interest she has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

Direct Variation 11-5 Example 3 Continued A. interest from CD and time Pre-Algebra 11-5 Direct Variation Example 3 Continued A. interest from CD and time interest from CD time = 17 1 interest from CD time = = 17 34 2 The second and third pairs of data result in a common ratio. In fact, all of the nonzero interest from CD to time ratios are equivalent to 17. = = = 17 interest from CD time = = 17 1 34 2 51 3 68 4 The variables are related by a constant ratio of 17 to 1, and (0, 0) is included. The equation of direct variation is y = 17x, where x is the time, y is the interest from the CD, and 17 is the constant of proportionality.

Direct Variation 11-5 Example 3 Continued Pre-Algebra 11-5 Direct Variation Example 3 Continued B. interest from money market and time interest from money market time = = 19 19 1 interest from money market time = =18.5 37 2 19 ≠ 18.5 If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Pre-Algebra 11-5 Direct Variation

Direct Variation 11-5 Helpful Hint Pre-Algebra 11-5 Direct Variation The graph of a direct-variation equation is always linear and always contains the point (0, 0). The variables x and y either increase together or decrease together. Helpful Hint

Direct Variation 11-5 Try This: Example 2B B. x is 28 when y is 14 Pre-Algebra 11-5 Direct Variation Try This: Example 2B B. x is 28 when y is 14 y = kx y varies directly with x. 14 = k  28 Substitute for x and y. = k 1 2 Solve for k. Substitute for k in the original equation. 1 2 y = x 1 2

Direct Variation 11-5 Try This: Example 2C C. y is 7 when x is 3 Pre-Algebra 11-5 Direct Variation Try This: Example 2C C. y is 7 when x is 3 y = kx y varies directly with x. 7 = k  3 Substitute for x and y. = k 7 3 Solve for k. Substitute for k in the original equation. 7 3 y = x 7 3

Direct Variation 11-5 Try This: Example 3 Pre-Algebra 11-5 Direct Variation Try This: Example 3 Mr. Lotkowski has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

Try This: Example 3 Continued Pre-Algebra 11-5 Direct Variation Try This: Example 3 Continued   Interest Interest from Time (mo) from CD ($) Money Market ($) 1 12 15 2 30 40 3 45 4 50

Try This: Example 3 Continued Pre-Algebra 11-5 Direct Variation Try This: Example 3 Continued A. interest from CD and time interest from CD time = 12 1 interest from CD time = = 15 30 2 The second and third pairs of data do not result in a common ratio. If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Try This: Example 3 Continued Pre-Algebra 11-5 Direct Variation Try This: Example 3 Continued B. interest from money market and time interest from money market time = = 15 15 1 interest from money market time = =20 40 2 15 ≠ 20 If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.