1. Linear Equations in Two Variables
3.3
Linear Equations in Two Variables
Slope-Intercept Form

2. Slope-Intercept Form
The slope-intercept form of the equation of a line with slope m and y-intercept (0, b) is
y = mx + b.
Slope y-intercept is (0, b).

3. EXAMPLE 1 Find an equation of the line with slope 2 and
y-intercept (0, –3).
m = 2
b = –3
Substitute these values into the slope-intercept form.
y = mx + b
y = 2x – 3

4. EXAMPLE 2 Graph the line, using the slope and y-intercept. x + 2y = –4
Write the equation in slope-intercept form by solving for y.
x + 2y = –4
2y = –x – Subtract x.
Divide by 2
y-intercept (0, –2)
Slope

5. continued Graph: 1. Plot the y-intercept. (0, -2) 2. The slope is
3. Using (-1/2), begin at (0,-2) and move 1 unit down and units right.
4. The line through these two points is the required graph.

6. Point-Slope Form
The point-slope form of the equation of a line with slope m passing through the point (x1, y1) is
y – y1 = m(x – x1).
Slope
Given point
If you do not like to deal with fractions, you can use your slope formula as well.

7. EXAMPLE 3
Find an equation of the line with slope 2/5 and passing through the point (3, –4).
Use the point-slope form with (x1, y1) = (3, –4) and m = 2/5.
Substitute
Multiply by 5.
Subtract 2x and 20.
Multiply by -1.

8. EXAMPLE 4
Find an equation of the line passing through the points (–2, 6) and (1, 4). Write the equation in standard form.
First find the slope by the slope formula.
Use either point as (x1, y1) in the point-slope form of the equation of a line.
Using the point (1, 4): x1 = 1 and y1 = 4

9. continued m = -2/3; x1 = 1 and y1 = 4
Substitute
Multiply by 3.
Add 2x and 12.
If the other point were used, the same equation would result.

10. Equations of Horizontal and Vertical Lines
The horizontal line through the point (a, b) has equation y = b.
The vertical line through the point (a, b) has equation
x = a.

11. EXAMPLE 5
Find an equation of the line passing through the point (–8, 3) and
a. parallel to the line 2x – 3y = 10;
b. perpendicular to the line 2x – 3y = 10.
Write each equation in slope-intercept form.
a. Find the slope of the line 2x – 3y = by solving for y.

12. Find an equation of the line passing through the point (–8, 3).
continued
The slope is
Parallel lines have the same slope. Use point slope form and the given point.
The fractions were not cleared because we want the equation in slope-intercept form instead of standard form.

13. Find an equation of the line passing through the point (–8, 3).
continued
b. Perpendicular lines. The slope is the negative reciprocal of
Use point slope form and the given point. m =

15. EXAMPLE 6
Suppose there is a flat rate of $0.20 plus a charge of $0.10 per minute to make a telephone call. Write an equation that gives the cost y for a call of x minutes.
y = $ $0.10x
or y = $0.10x

16. EXAMPLE 7
The percentage of the U.S. population 25 years and older with at least a high school diploma is shown in the table for selected years. Find an equation that models the data, using x = 0 to represent 1940, x = 10 to represent 1950, and so on.
Year
Percent
1940
24.5
1950
34.3
1960
41.4
1970
52.3
1980
66.5
1990
75.2
2000
80.4

17. continued
Choose two data points and find the slope. Use 1940 and 2000.
The y-intercept is (0, 24.5).
The equation is:
y = 0.93x
Selecting two different ordered
pairs will lead to a different equation.
Year
Percent
1940
24.5
1950
34.3
1960
41.4
1970
52.3
1980
66.5
1990
75.2
2000
80.4

18. EXAMPLE 8
Use the ordered pairs (11, 164) and (13, 203) to find an equation that models the data in the graph below.
Use the point-slope form with (11, 164).

19. Homework
pg 186 # 4, 6-90 m6, 97, 98
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