College Algebra Lecture Notes
Section 2.2
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Section 2.2: Graphs of Linear Equations
Big Idea: The graph of a line is always represented by the equation ax  by  c .
Big Skill: .You should be able to graph a line given its equation, and determine when lines are
parallel or perpendicular.
A. The Graph of a Linear Equation
Linear Equations
A linear equation is one that can be written in the form ax  by  c where a and b are not both
zero.
 Why can’t a and b both be zero?
 The graph of a linear equation is always a line. That’s why it’s called linear. See
problem #1.
 Since there is only one line between any two points in a plane, you need only plot two
points to graph any linear equation.
 Notice that most lines, like in problems#1 and #2, cross both the x-axis and the y-axis.
 The values at which the line crosses the axes are called the x-intercept and the y-intercept.
 Intercepts are convenient to use as the two points for graphing any line:
The Intercept Method
1. Substitute x = 0 and solve for y. This gives the y-intercept (0, y).
2. Substitute y = 0 and solve for x. This gives the x-intercept (x, 0).
3. Plot the intercepts and draw a straight line between them.
Practice:
1. Graph the equation 2  x  3  3  y  7  by plotting points.
College Algebra Lecture Notes
Section 2.2
Page 2 of 5
2. Graph the equation x  y  2 by plotting two points.
3. Graph the equation 4 x  3 y  7  0 using the Intercept Method.
B. The Slope of a Line
The Slope Formula
Given two points P1   x1 , y1  and P2   x2 , y2  , the slope of any nonvertical line through P1 and
y2  y1
, where x1  x2.
x2  x1
Be consistent when plugging in the x’s and y’s…
Be careful to avoid sign errors when subtracting a negative.
When m > 0, then the line between the points increases from left to right.
When m < 0, then the line between the points decreases from left to right.
The slope measures the rate of change of the y quantity with respect to the x quantity.
P2 is: m 
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College Algebra Lecture Notes
Section 2.2
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Practice:
4. Compute the slope between the points (-3, -2) and (4, 6) and draw the line between them.
Do the same for the points (-5, 5) and (8, -7).
5. A student lives 45 miles from MATC. He leaves home at 8AM and arrives at campus at
8:45 AM. Write an equation that models his distance from MATC as function of time
and graph it.
C. Horizontal and Vertical Lines
Horizontal Lines
The equation of a horizontal line is y = k, where (0, k) is the y-intercept.
The Slope of a Horizontal Line
The slope of any horizontal line is zero.
Vertical Lines
The equation of a vertical line is x = h, where (h, 0) is the x-intercept.
The Slope of a Vertical Line
The slope of any vertical line is undefined.
College Algebra Lecture Notes
Section 2.2
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Practice:
6. Graph x = 3 and y = -7 below.
D. Parallel and Perpendicular Lines
Parallel Lines
Given that L1 and L2 are distinct non-vertical lines with slopes of m1 and m2 respectively, then:
 If m1 = m2, then the lines are parallel
 If L1 and L2 are parallel, then m1 = m2.
 Shortcut for the above two points: L1 and L2 are parallel if and only if m1 = m2.
 Shortcut for saying L1 and L2 are parallel: L1 || L2
 Any two vertical lines are also parallel.
Perpendicular Lines
Given that L1 and L2 are distinct non-vertical lines with slopes of m1 and m2 respectively, then:
 If m1  m2, = -1 then the lines are perpendicular.
 If L1 and L2 are perpendicular, then m1  m2, = -1.
 Shortcut: L1 and L2 are perpendicular if and only if m1  m2, = -1.
 Shortcut for saying L1 and L2 are perpendicular: L1  L2
 Any vertical line is perpendicular to any horizontal line.
Practice:
7. Determine if the points (5, 1), (3, -2), and (-3, 2) form the vertices of a right triangle, and
then find a pair of points parallel to one of the sides.
College Algebra Lecture Notes
D. Applications of Linear Equations
Section 2.2
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