This document discusses how to solve systems of equations and inequalities by graphing. It explains that systems contain two or more equations or inequalities to be solved simultaneously. Graphing the functions allows identification of intersection points, which are the solutions. Systems can have one solution, no solution, or infinitely many solutions depending on whether the graphs intersect, are parallel, or coincide. The steps are to graph each function on the same plane and identify intersection points of solutions. Systems of inequalities are graphed similarly, with the region of overlap indicating the solution set. Examples demonstrate solving various systems of equations and inequalities through graphing.

1. SOLVING SYSTEMS OF EQUATIONS
AND INEQUALITIES BY GRAPHING

2. SYSTEMS OF EQUATIONS

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).

3. 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.

4. 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.

5. EXAMPLE 1

Let’s solve the system
below by graphing:

Graph each function on
the same coordinate
plane:

6. 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!

7. 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.

8. 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).

9. 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.

10. 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.

11. 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 xaxis.

The second equation is a line.

Now, graph.

12. 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.

13. 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.

14. 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

15. SYSTEMS OF INEQUALITIES CONTINUED

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

16. 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.

17. 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.

18. EXAMPLE 2

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.

19. EXAMPLE 3

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.




20. UP NEXT…
 In
Lessons 4 and 5, you will study a
real-world application of solving
systems of linear equations and
inequalities!