HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Hawkes Learning Systems:
College Algebra
Section 3.2: Linear Equations in Two Variables
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Objectives
o Recognizing linear equations in two variables.
o Intercepts of the coordinate axes.
o Horizontal and vertical lines.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Recognizing Linear Equations in Two Variables
Linear Equations in Two Variables
A linear equation in two variables, say the variables x
and y, is an equation that can be written in the form
ax  by  c
where a , b , and c are constants and a and b are not
both zero. This form of such an equation is called the
standard form.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 1: Linear Equations
Determine if the equation is a linear equation.
2 x  3y  5x  x  2 y  1
2x  6 y  5x  x  2 y  1
3 x  6 y  x  2 y  1
3 x  6 y  x  2 y  1
4 x  4 y  1
The equation is linear.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 2: Linear Equations
Determine if the equation is a linear equation.
4x  3 x  5  y   7 x
4 x  3 x  15  3 y  7 x
7 x  7 x  3 y   15
 3 y   15
y5
The equation is linear.
Note: One of the variables is absent from the resulting
equation, but since the coefficient of y is non-zero,
this equation is still linear.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 3: Linear Equations
Determine if the equation is a linear equation.
x3
 y
4
x

x
8
3
 y
x
4
4
8
2x
x
3
8

 y
8
4
3x
3
8
 y 
4
The equation is linear.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 4: Linear Equations
Determine if the equation is a linear equation.
2 x   x  5  6 y  2 3 y  2   x
2x  x  5  6 y  6 y  4  x
x  x  6y  6y  4 5
09
The equation is not linear.
Note: The equation is not linear because the coefficients
of x and y are both 0. Moreover, this equation has
no solution: no values for x and y result in a true
statement.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 5: Linear Equations
Determine if the equations are linear equations.
a.
6 y  3x  7 y
3
The equation is not linear.
Note: The presence of the
cubed term in this already
simplified equation makes
it clearly not linear.
4 x   2 x  3  8 y
2
2
b.
4 x  4 x  12 x  9  8 y
2
2
12 x  8 y  9
The equation is linear.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Intercepts of the Coordinate Axes
If the straight line whose points constitute the solution
set crosses the horizontal and vertical axes in two
distinct points, knowing the coordinates of these two
points is sufficient to graph the complete solution.
y-axis
 x1 , y1 
 x2 , y2 
x-axis
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Intercepts of the Coordinate Axes
o For an equation in the two variables x and y, it is
natural to call the point where the graph crosses
the x-axis the x-intercept, and the point where it
crosses the y-axis the y-intercept.
o The y-coordinate of the x-intercept is 0, and the x coordinate of the y-intercept is 0.
y-axis
y-intercept
x-intercept
x-axis
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 6: Intercepts
Find the x - and y -intercepts of the equation and graph.
3 x  4 y  12
3  0   4 y  12
y  3
y -intercept:  0,  3 
3 x  4  0   12
x4
x -intercept:  4, 0 
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 7: Intercepts
Find the x - and y -intercepts of the equation and graph.
4 x  3  x   2 y  7
5 x  2 y  10
5  0   2 y  10
y5
y -intercept:  0, 5 
5 x  2  0   10
x2
x -intercept:  2, 0 
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Horizontal and Vertical Lines
o A given linear equation may not have one of the two
types of intercepts. This can only happen when the
graph of the equation is a horizontal or vertical line.
o In the absence of other information, it is impossible
to know if the solution of a linear equation missing
one of the two variables consists of a point on the
real number line, or a line in the Cartesian plane.
o You must rely on the context of the problem to know
how many variables should be considered.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Horizontal and Vertical Lines
Consider an equation of the form a x  c . The variable y is
absent, so any value for this variable will suffice, as long as
we pair it with x 
c
a
. Thinking of the solution set as a set
of ordered pairs, the solution consists of ordered pairs with
a fixed first coordinate and arbitrary second coordinate.
This describes a vertical line with an x-intercept of
c
a
.
Similarly, the equation by  c represents a horizontal line
with y -intercept equal to
c
b
.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 8: Horizontal and Vertical Lines
Graph the following equation.
3x  2  x  7   2 y  5x
5 x  5 x  14  2 y  0
 2 y   14
y7
Note: The graph of the
equation is the
horizontal line consisting
of all those ordered
pairs whose
y-coordinate is 7.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 9: Horizontal and Vertical Lines
Graph the following equation.
5x  0
x0
Note: the graph of this equation is the y -axis, as all the
ordered pairs on the y -axis have an x -coordinate of 0.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 10: Horizontal and Vertical Lines
Graph the following equation.
2x  2  3
5
x
2
Note: This
equation is a
vertical line, which
passes through
5
5/on
the x-axis.
2