Algebra 1: Functions and Linear Equations : Section Four

Slope-Intercept Form

When Carmen started working in 2002, she kept track of her earnings each year. Here is a table of her earnings to 2005, rounded to the nearest thousand.

x(year)

y(annual earnings,$thousands)

2002

18,000

2003

20,000

2004

22,000

2005

24,000

You can use this information to find the equation of a line in slope-intercept form. Before getting started, let’s review slope-intercept form.

Slope-Intercept Form

y = mx + b

m is the slope

b is the y-intercept

Example 1: Write an Equation Using a Slope and One Point
In the example above, one point on the line is (2003, 20,000) and the slope is 2,000. Write an equation in slope-intercept form for the line that passes through the point with slope 2000.

Start by substituting known quantities in the slope-intercept equation.
y = mx + b
20,000 = 2000(2003) + b Substitute x = 2003, y = 20,000, and m = 2000
20,000 = 4,006,000 + b Simplify
20,000 – 4,006,000 = b Subtraction property
-3,986,000 = b Simplify

Now, substitute the values for m and b into the slope-intercept equation.
y = mx + b
y = 2000x – 3,986,000
Therefore, the equation which represents Carmen’s annual earnings is
y = 2000x – 3,986,000.

Check: One way to check this problem is to substitute a different data point and verify that the equation balances.
y = 2000x – 3,986,000.
18,000 = 2000(2002) – 3,986,000
18,000 = 18,000 √

Next, use this equation to predict future earnings for Carmen. When you use a linear equation to predict values beyond the data range, you are using linear extrapolation.

Example 2: Linear Extrapolation
Assuming that Carmen’s annual earnings will continue to increase at the same rate, find Carmen’s earnings in 2010, using the equation y = 2000x – 3,986,000.

In 2010:
y = 2000x – 3,986,000
y = 2000(2010) – 3,986,000 = $34,000

Example 3: Find an Equation Using a Point and Slope
If a line passes through the point (2, 6) with a slope of - ¾, what is its slope-intercept equation?

y = mx + b Slope-Intercept Form
6 = - ¾ (2) + b Substitute x = 2, y = 6, and m = - ¾
6 = - 3/2 + b Simplify
6 + 3/2 = b Subtraction property
15/2 = b Simplify
Note: If you use the decimal form, it must be terminating and you should change the slope to a decimal also. Do not use decimals if you would needto round them, unless the problem calls for an estimate, such as a real lifeproblem. Leave all as fractions in this case.

Example 4: Find an Equation Using Two Points
Find the equation of the line which passes through (4, 5) and (-1, 3).

Since you know how to find the equation of a line using a slope and a point, find the slope of the line formed by the points.

Next, use this slope and one of the points.
m = 2/5 and (4, 5)
y = mx + b

y = mx + b
y = 2/5 x + 17/5 or y = 0.4x + 3.4

Check:
Substitute the point not chosen earlier, (-1, 3).
y = 0.4x + 3.4
3 = 0.4(-1) + 3.4
3 = 3 √

To Write an Equation in Slope-Intercept Form

Using the Slope and One Point

Using Two Points

1. Substitute the values of m, x,
and y into y = mx + b.

1. Find the slope

2. Solve for b.

2. Choose one of the points

3. Substitute the values of m
and b into y = mx + b.

3. Follow the steps for using the slope
and one point.

Example 5: Population of California
According to the U.S. Census Bureau, these are the estimated population figures for California in the years shown in the table.

Use the estimates for 1990 and 2005 to write a linear equation that represents the data.

Start with the ordered pairs that represent 1990 and 2005,
(1990, 29.8) and (2005, 36.1).
Note that 29.8 and 36.1 represent millions.

Using the technique outlined above, start by finding the slope.

Use m = 0.42 and the point (2005, 36.1).
y = mx + b
36.1 = 0.42(2005) + b
36.1 = 842.1 + b
-806 = b

y = mx + b
y = 0.42x – 806

Check: Substitute the point not chosen, (1990, 29.8).
y = 0.42x – 806
29.8 = 0.42(1990) – 806
29.8 = 29.8 √

Example 6: Using the Two Intercepts
Find an equation of the line with an x-intercept of -6 and a y-intercept of -2.

The two points are (-6, 0) and (0, -2).

b is the y-intercept, so it is -2.
y = -1/3 x – 2

Check: Use the x-intercept.
0 = -1/3 (-6) – 2
0 = 2 – 2
0 = 0 √

Quick Practice

Use linear extrapolation to predict the population of California in 2012 from the equation y = 0.42 x – 806.

y = 0.42 (2012) – 806 = 39.04 million

Write an equation of the line that passes through (6, -2) with slope -3.

y = -3x + 16

Write an equation of the line that passes through (-3, -1) with slope 2/3.

y = 2/3 x + 1

Write an equation of the line that passes through (4, 1) and (6, -2).

y = - 3/2 x + 7 or y = -1.5 x + 7

Write an equation of the line that passes through (-5, 2) and (6, 2).

y = 0x + 2 or y = 2

Write an equation of the line that has these intercepts: x-intercept of 4 and y-intercept 3.

y = - ¾ x + 3 or y = -0.75x + 3

These are the cumulative totals of Harry’s earnings in his job so far. He was paid for a training day and for the first four days of work last week. These are days five through eight.

x (Day)

5

6

7

8

y (Cumulative Pay)

$395

$463

$531

$599

Use any two ordered pairs from the table to find the linear equation that represents Harry’s cumulative pay.

y = 68x + 55

What do the slope and y-intercept represent?

The slope represents $68 earned per day and the y-intercept represents $55 earned on the first day.

Use linear extrapolation with the equation you found in number 7 above to predict Harry’s cumulative pay on day 10.

$735

HW 7 (13 points)

Go to the left menu bar and click on the homework. You may do this more than one time to improve your score. Good luck!