Math 150 – Fall 2015
Section 3C
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Section 3C – Linear Equations in Two Variables
Linear Equations in Two Variables
Definition. Any equation that can be written in the form Ax + B = C, where A and
B are not both 0, is called a linear equation. This equation is called the general
form of the equation of a straight line.
Example 1. Graph the following lines by plotting points. Make sure to add the x and
y intercepts to the graph.
(a) 3x − 2y = −8
(b) 3y − 7 = 0.
Slopes of Lines
Definition. The slope of a line, m, is the ratio of the change in y to the change in x.
m=
change in y (4y)
rise
=
run
change in x (4x)
The slope of the line through points P1 (x1 , y1 ) and P2 (x2 , y2 ) is
m=
y2 − y1
x2 − x1
Example 2. Plot each pair of points on paper and find the slope of the line:
(a) (−2, −3) and (4, −1)
(b) (−3, 5) and (5, 5)
Math 150 – Fall 2015
Section 3C
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(c) (2, 5) and (2, −2)
(d) (−4, 4) and (0, −3)
Example 3. Use the results to complete each sentence.
(a) The line is increasing (goes uphill from left to right) when the slope is
(b) The line is decreasing (goes downhill from left to right) when the slope is
(c) The line is horizontal when the slope is
(d) The line is vertical when the slope is
.
.
Slope-Intercept Form
Theorem. A line is in slope-intercept form if it is written as y = mx + b. For this
form, we have the following facts:
• The y-intercept is b so the line intersects the y-axis at the point (0, b).
• The coefficient m of x is the slope of the line.
To graph a line in slope-intercept form y = mx + b:
• Plot the point (0, b)
• Apply the slope m to find a second point on the line.
• Draw a line through the two points.
Example 4. Graph y = −2x + 4
.
.
Math 150 – Fall 2015
Section 3C
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Example 5. Find the slope and y-intercept of the line −2x + 4y = 7 and then graph
it.
Example 6. Write the equation of the line with slope
−3
2
through the point (0, −2).
Point-Slope Form
Definition. The equation of a line with slope m that passes through the point (x1 , y1 )
is
y − y1 = m(x − x1 ).
This is called the point-slope form of the equation of a line.
Example 7. Write the slope-intercept form of the line that has slope m =
through the point (4, −2). Also, write the equation in general form.
2
3
and goes
Example 8. Write the slope-intercept form of the line that goes through the points
(5, 4) and (−3, −3). Also, write the equation in general form.
Horizontal Lines and Vertical Lines
Note. Earlier we noted that horizontal lines have slope m = 0. Therefore, a horizontal
line through the point (a, b) is just the line y = b. Vertical lines have undefined slope
and the equation of a vertical line through the point (a, b) is x = a.
Example 9. Find the equations of the vertical line and the horizontal line that goes
through the point (7, −3).
Math 150 – Fall 2015
Section 3C
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Parallel and Perpendicular Lines
Definition. Two lines are parallel if they have the same slope. Two lines are perpendicular if the slopes are negative reciprocals of each other, i.e., if the slope of one line
is ab , then the slope of the other line is − ab .
Example 10. Find the slope of any line that is parallel to and perpendicular to the
line through the points A(2, 4) and B(5, 6).
Example 11. Find the line that goes through the point (−2, 5) and is parallel to the
line 3x + 5y = 7. Also, find the line that goes through (−2, 5), but is perpendicular to
3x + 5y = 7.
Example 12. Find the slope-intercept form of the perpendicular bisector of the line
segment connecting A(4, 7) and B(8, −2).
Example 13. If the line through A(3, −5) and B(−2, y2 ) is perpendicular to the line
−5x + 4y = 2, find y2 .