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8-3-1

Chapter 1

Section 8-3Equations of Lines and Linear Models

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8-3-2

Equations of Lines and Linear Models

• Point-Slope Form

• Slope-Intercept Form

• Summary of Forms and Linear Equations

• Linear Models

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8-3-3

Point-Slope Form

The equation of the line through (x1, y1) with slope m is written in point-slope form as

1 1( ).y y m x x

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8-3-4

Example: Finding an Equation Given the Slope and a Point

Find the standard form of an equation of the line with slope 1/3, passing through the point (–3, 2).

Solution

1 1( )y y m x x 1

2 ( ( 3))3

y x

3 6 3y x

3 9x y Standard form

Multiply by 3

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8-3-5

Example: Finding an Equation Given Two Points

Find the standard form of an equation of the line with passing through the points (2, 1) and (–1, 3).

Solution3 1 2

1 2 3m

21 ( 2)

3y x

2 3 7x y Standard form

Find the slope.

Use either point in the form.

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8-3-6

Slope-Intercept Form

The equation of the line with slope m and y-intercept (0, b) is written in slope-intercept form as

.y mx b

Slope y-intercept

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8-3-7

Example: Graphing Using Slope and the y-Intercept

y

x

Graph the line with equation

Solution

32.

2y x

Plot the intercept (0, –2) and use the slope: rise 3, run 2.

rise 3

run 2

(0, –2)

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8-3-8

Summary of Forms of Linear Equations

y mx b

y b

x a

1 1( )y y m x x

Ax By C Standard form

Horizontal line

Vertical line

Slope-intercept form

Point-Slope form

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8-3-9

Linear Models

Earlier examples gave equations that described real data. The process of writing an equation to fit a graph is called curve-fitting. The next example illustrates this concept for a straight line. The resulting equation is called a linear model.

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8-3-10

Example: Modeling Costs

Estimates for Medical costs (in billions of dollars) are shown below.

Year Cost (in billions)

2000 225

2001 243

2002 261

2003 279

2004 297

a) Find a linear equation that models the data.

b) Use the model to predict the costs in 2010.

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8-3-11

Example: Medical Costs

SolutionLet x = 0 correspond to 2000, x = 1 correspond to 2001, and so on. We can express the data as ordered pairs:

(0, 225), (1, 243), (2, 261), (3, 279), and (4, 297).

a) To find a linear equation through the data we choose two points to get the slope. Using (0, 225) and (3, 279): 279 225 54

18.3 0 3

m

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8-3-12

Example: Medical Costs

Solution (continued)

Now since we have the y-intercept (0, 225) we have the equation

b) The value x = 10 corresponds to the year 2010. When x = 10,

18 225.y x

18(10) 225 405y

The model predicts that the costs will be $405 billion in 2010.