Please get out your completed 8.2B Graphing
Worksheet 2 and pass it to the center aisle to
be collected by the TA.
Today’s quiz on 8.2 A Graphing Worksheet 1 will
be given at the end of class. You will have 12
minutes to complete this quiz, which will consist
of one graphing problem from that worksheet.
Your graded Worksheet 2 will be returned at the
start of class tomorrow, and you will have a quiz
on that material at the end of class.
Teachers: After you return Worksheet 1, you might want to spend a few minutes going over any common errors you noticed while reviewing your students’ worksheets
before you start the lecture for section 4.1. Please remember that you’ll need to wind up your lecture with about 15 minutes remaining so that students will have time to open
and complete the 12-minute quiz. This quiz has a special answer sheet with a graphing grid, so please make sure your TA hands out the correct answer sheet for today’s quiz.
2
Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
Section 4.1
Solving Systems of Equations in
Two Variables by Graphing
• A system of linear equations consists of two
•
•
or more linear equations.
This section focuses on only two equations at
a time.
The solution of a system of linear equations
in two variables is any ordered pair that
solves both of the linear equations.
What does this look like on a graph?
The SOLUTION to a system of two linear
equations is the intersection (if any) of the two
lines.
There are only three possible solution scenarios:
1. The lines intersect in a single point (so the answer is
one ordered pair).
2. The lines don’t intersect at all, i.e. they are parallel
(so the answer is “no solution”.)
3. The two lines are identical, i.e. coincident, so there
are infinitely many solutions (all of the points that
fall on that line.)
To be a SOLUTION of a system of
equations, an ordered pair must result in
true statements for BOTH equations
when the values for x & y are plugged into
them. If either one (or both) gives a false
statement, the ordered pair is NOT a
solution of the system.
Example
Determine whether the given point is a solution of
the following system.
point: (-3, 1)
system: x – y = -4 and 2x + 10y = 4
• Plug the values into the equations.
First equation: -3 – 1 = -4 true
Second equation: 2(-3) + 10(1) = -6 + 10 = 4
true
• Since the point (-3, 1) produces a true statement
in both equations, it is a solution.
Example
Determine whether the given point is a solution of
the following system
point: (4, 2)
system: 2x – 5y = -2 and 3x + 4y = 4
• Plug the values into the equations
First equation: 2•4 - 5•2 = 8 – 10 = -2
true
Second equation: 3•4 + 4•2 = 12 + 8 = 20  4
false
• Since the point (4, 2) produces a true statement
in only one equation, it is NOT a solution.
Problem from today’s homework:
(try this one in your notebook)
x = 3 and y = 5 works in the first
equation, but not in the second one.
•
•
•
Since a solution of a system of equations is a
solution common to both equations, it would also
be a point common to the graphs of both
equations.
One way to find the solution of a system of 2
linear equations is to graph the equations and see
where the lines intersect.
You can use any of the techniques from Chapter 3
to graph the two lines (e.g. solving each equation for y and
using the slope and intercept, or making a table of x- and y-values for
each equation and plotting the ordered pairs.)
Graphing is the first of three methods for solving
systems of equations that we will be studying in
this chapter. The other two methods we will be
using are:
• Substitution method (Section 4.2)
• Addition or elimination method (Section 4.3)
Note: Graph Paper
Click on the “Announcements” button to find a site that
allows you to print free graph paper.
• If you want to be able to draw accurate graphs but you
don't want to buy a whole pack of graph paper for one
assignment, go to this web site and print a couple pages of
graph paper for free. (You don’t have to do this – graphing
by hand on plain paper is fine, but sometimes it’s easier to
see the solutions if you can plot your points carefully on
real graph paper instead of a hand-drawn graph grid.)
http://www.printfree.com/Office_forms/GraphPaper2.htm
Example:
Solve the following system of equations by graphing:
y
2x – y = 6 and
x + 3y = 10
First, graph 2x – y = 6.
• Solving for y gives y = 2x – 6.
• Plot the y-intercept of -6, then use the
slope of 2 to go up two, over one.
(4, 2)
(1, 3)
x
Second, graph x + 3y = 10.
• Solving for y gives y = -1/3x + 10/3.
• The y-intercept is a fraction, so let x = 1;
then y = -1/3 + 10/3 = 9/ 3 = 3.
• Plot (1,3), then use the slope of -1/3 to go
down one, over three.
(1, -4),
(0, -6)
The lines APPEAR to intersect at (4, 2).
Example (cont.)
Although the solution to the system of
equations appears to be (4, 2), you still need
to check the answer by substituting x = 4 and
y = 2 into the two equations.
First equation,
2(4) – 2 = 8 – 2 = 6
true
Second equation,
4 + 3(2) = 4 + 6 = 10 true
The point (4, 2) checks, so it is the solution of
the system.
Problem from today’s homework:
(try this one in your notebook)
Solution:
• Graph the 2 lines.
• They appear to intersect at (3,3)
• Now check x = 3, y = 3 back
into BOTH equations to make
sure they both give true
statements.
Solve the following system of
equations by graphing.
Example
y
-x + 3y = 6 and
3x – 9y = 9
First, graph -x + 3y = 6.
• Solving for y gives y = 1/3x + 2.
• Plot the y-intercept of 2, then use the
slope of 1/3 to go up one, over three.
Next, graph 3x - 9y = 9.
• Solving for y gives y = 1/3x - 1.
• Plot the y-intercept of -1, then use the
slope of 1/3 to go up one, over three.
The lines APPEAR to be parallel.
(3,3)
(0, 2)
(3, 0)
(0, -1)
x
Example (cont.)
•
•
Although the lines appear to be parallel, you still
need to check that they have the same slope. You
can do this by solving both equations for y, as we
did on the previous slide.
1
Both lines have a slope of 3 ; since they have different
y-intercepts they are parallel and do not intersect.
Hence, there is no solution to the system.
Example
y
Solve the
following system
of equations by
graphing.
x = 3y – 1 and
2x – 6y = -2
(5, 2)
(-1, 0)
(-4, -1)
(7, -2)
First, graph x = 3y – 1.
Second, graph 2x – 6y = -2.
The lines APPEAR to be identical.
(2, 1)
x
Example (cont.)
•
Although the lines appear to be identical, you still
need to check that they are identical equations.
You can do this by solving for y.
First equation,
x = 3y – 1
3y = x + 1
y= 1x+
3
(add 1 to both sides)
1
(divide both sides by 3)
3
Second equation,
2x – 6y = -2
-6y = -2x – 2
y= 1x+ 1
•
3
(subtract 2x from both sides)
(divide both sides by -6)
3
The two equations are identical, so the graphs must
be identical. There are an infinite number of solutions
to the system (all the points on the line y = 1/3 x + 1/3).
REMINDER: At the next class, you will get your graded
Worksheet 2 back, and you’ll have a quiz on those graphs
at the end of the class session.
HW 4.1 on today’s material is due at the start of the next
class session. That material will be covered on the quiz
two class sessions from today, along with material from
HW 4.2.
Lab hours:
Mondays through Thursdays
8:00 a.m. to 7:30 p.m.
Please open your laptops, log in to the MyMathLab
course web site, and open Quiz 8.2 Worksheet 1.
• You will have 12 minutes to complete
this quiz.
• No notes allowed (including graded worksheet.)
• The answer sheet has:
• a table for your x-y values
(minimum of eight ordered pairs)
• a graph grid for plotting the points
• spaces for writing the x- and yintercepts (write as ordered pairs,
or write NONE if there is no intercept)
• spaces for writing the domain and range
in interval notation
• a table of square roots in case you need
them for your graph
• a spare graph grid in case you make a
mistake on the first one and don’t want
to erase.