Problems of the Day1.) Find the slope of the line that contains (–9, 8) and (5, –4).

2.)Find the value of r so that the line through (10, -3) and (r, 6) has a slope of -3/2.

7

6

14

12m

4

123

18303

)9(2)10(310

9

2

310

)3(6

2

3

r

r

r

rr

r

Chapter 5.2

Slope and Direct Variation

A recipe for paella calls for 1 cup of rice to make 5 servings. In other words, a chef needs 1 cup of rice for every 5 servings.

The equation y = 5x describes this relationship. In this relationship, the number of servings varies directly with the number of cups of rice.

A direct variation is a special type of linear relationship that can be written in the form y = kx, where k is a nonzero constant called the constant of variation.

Name the constant variation for the equation. Then find the slope of the line that passes through each pair of points.

The constant variation is -1/2

The slope is, 2

1

40

20

The equation y = − ½x is in y = kx form.

Ex. 1

In this example, the constant variation and slope are both equal to 4/3.

In this example the SLOPE

of the line is positive.

In the previous slide, the

slope of the line was negative.

Ex. 2

Graph the direct variation equation y = 2x

1.) First coordinate is (0, 0)…when you plug in x = 0, y = 0.

2.) Write the slope as a fraction. Remember slope is rise over run.

Slope =

3.) From (0, 0) we “rise” 2 units and “run” 1 unit. Draw a point.

4.) Draw a line through the 2 points.

1

2

Ex. 3

Graph y = -4x

1.) First coordinate is (0, 0)…when you plug in x = 0, y = 0.

2.) Write the slope as a fraction. Remember slope is rise over run.

Slope =

3.) From (0, 0) we “rise” -4 which means go down 4 units and “run” 1 unit. Draw a point.

4.) Draw a line through the 2 points.

1

4

Ex. 4

Write and Solve a Direct Variation Equation

Suppose y varies directly as x, and y = 28 and x = 7. Write a direct variation equation (y = kx) that relates x to y.

We have to find the value of k first using the equation y = kx

We know y = 28 and x = 7. Plug in and solve!! 28 = k ∙ 7

k = 4

So the direct variation equation is y = 4x.

Ex. 5

Example 5 continued…

Use the direct variation equation to find x when y = 52.

Remember the equation we found was y = 4x

If y = 52, plug in and solve!! 52 = 4 ∙ x

So x = 13, when y = 52

The value of y varies directly with x, and y = 3, when x = 9. Write the direct variation equation (y = kx), then use this equation to find y when x = 21.

y = kx

3 = k(9)

3

1

9

3k

xy3

1

)21(3

1y

7y

Ex. 6

Direct Variation Equation

A local fast food restaurant takes in $9000 in 4 hours.

a.) Write a direct variation equation for the amount of money taken in any time.

Total Income = Income per hour • Number of hours

$9000 = k • 4

9000 = 4k

k = 2250

Direct variation equation: y = 2250x,where y is total income and x is number of hours

Ex. 7

b.) How many hours would it take for the restaurant to earn $20,250?

y = 2250x

20,250 = 2250x

x = 9 hours

Remember, the direct variation equation is y = 2250x,where y is total income and x is number of hours

Assignment Study Guide 5-2 (In-Class) Skills Practice Worksheet 5-2

(Homework)