Download Solving systems of equations and inequalities by graphing

SOLVING SYSTEMS OF
EQUATIONS AND INEQUALITIES
BY GRAPHING
SYSTEMS OF EQUATIONS

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Remember that a system of equations is a group
of two or more equations that we solve at the
same time
A point is a solution of the system if it works
when substituted into each equation. For
example, the solution to the system above is (2,0).
REVIEW OF GRAPHS OF SYSTEMS OF
LINEAR EQUATIONS

When working with two equations in two
variables, there are three possibilities for their
graphs:
The lines can
intersect and
have one solution
(x, y).
The lines can be
parallel and have
no solution.
The lines can
coincide and have
infinitely many
solutions.
BUT NOW…


We want to start working with systems that
don’t just have linear equations.
We will still graph our functions and look for the
point(s) of intersection when we want to solve our
systems.
EXAMPLE 1

Let’s solve the system
below by graphing:

Graph each function
on the same
coordinate plane:
EXAMPLE 1 CONTINUED

Look at the graph and
identify the points of
intersection:
There are two points of
intersection, so our
system has two
solutions:
(-1, 2) and (1, 2)
You can substitute both
points into your
equations and get true
statements. This is an
easy way to check your
work!
TO SOLVE USING YOUR CALCULATOR


Put your equations in y =. abs( can be found by
pressing 2nd 0, and choosing the first option.
Graph to see the number of solutions.
TO SOLVE USING YOUR
CALCULATOR…CONTINUED


To find the first point of intersection, press 2nd
TRACE, and choose #5 (intersect). Move your
cursor to the left of the first intersection and
press enter. Move to the right and press enter.
Then press enter a third time to see the
coordinates:
Repeat the process to find the second solution at
(1, 2).
EXAMPLE 2


Let’s solve:
First, recognize that the first equation is an
absolute value graph (a V) that has been shifted
right 2 units and down 1 unit.

Then, solve the second equation for y: y = x + 1.

Finally, graph.
EXAMPLE 2 CONTINUED



The graphs intersect
ONCE.
The only solution to
the system is (0, 1).
Notice that you can
substitute your point
into both equations
and get a true
statement.
EXAMPLE 3


Let’s solve:
First, solve the first equation for y to get
. Then, recognize that this is an
absolute value graph (a V) that has been shifted
left 2 units, down 2 units, and reflected across
the x-axis.

The second equation is a line.

Now, graph.
EXAMPLE 3 CONTINUED



The graphs don’t
intersect.
The solution is that
there is no solution.
This means there is
NO point that exists
that would give you a
true statement for
both equations.
SUMMARY OF STEPS



Graph each function in your system. It would be
most helpful if you solve for y in each case.
Identify the point(s) of intersection of the graphs
of your functions.
State your solution(s). Check them by
substituting back into your system of equations.
SYSTEMS OF INEQUALITIES


Remember that a system of inequalities is a
group of two or more equations that we solve at
the same time:
Here’s a review of what the symbols tell us to do:
>:
 <:

:

:

dashed line, shaded above boundary line
dashed line, shaded below boundary line
solid line, shaded above boundary line
solid line, shaded above boundary line
SYSTEMS OF INEQUALITIES CONTINUED
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We will graph each boundary line just as we did
before, and we will put each of them on the same
coordinate plane.
Where the shaded regions all overlap will
represent the solution of our system—meaning
that any point from the shared region will
produce a true solution when substituted into all
of the inequalities in our system
EXAMPLE 1

Let’s solve the system
below by graphing:
The first will be a
dashed line shaded
above. (in red)
 The second will be a
solid line shaded
below. (in blue)


Graph each inequality
on the same
coordinate plane. The
area where they
overlap is the
solution.
EXAMPLE 1 CONTINUED


The region where both
shaded areas overlap
represents the solution
to our system. Notice
the region occurs in both
Quadrant II and in
Quadrant III.
Any point chosen from
this area will produce
true statements when
substituted into both
inequalities.
EXAMPLE 2
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Solve the system by
graphing:
The first is an absolute
value function; use a solid
line and shade above. (in
red)
The second is a horizontal
line; use a dashed line and
shade below. (in blue)
Since the shaded regions
don’t overlap, this system
has no solution.
EXAMPLE 3
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
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Solve the system by
graphing:
The first is a vertical line.
Use a solid line and
shade to the right.
The second is a vertical
line. Use a solid line and
shade to the left.
The third is a diagonal
line. Solve for y. Then
use a solid line and shade
below.
The solution region is
shaded the darkest.
UP NEXT…
 In
Lessons 4 and 5, you will study a
real-world application of solving
systems of linear equations and
inequalities!