(Sections 7.1-7.3: Systems of Equations)
7.01
CHAPTER 7: SYSTEMS AND INEQUALITIES
SECTIONS 7.1-7.3: SYSTEMS OF EQUATIONS
PART A: INTRO
A solution to a system of equations must satisfy
all
of the equations in the system.
In your Algebra courses, you should have learned methods for solving systems of linear
equations, such as:
A
+
B
=
1
A
−
4
B
=
11
⎧
⎨
⎩
We will solve this system using both the Substitution Method and the
Addition / Elimination Method in
Section 7.4
on Partial Fractions.
In some cases, these methods can be extended to nonlinear systems, in which at least one
of the equations is nonlinear.

(Sections 7.1-7.3: Systems of Equations)
7.02
PART B: THE SUBSTITUTION METHOD
See
Example 1 on p.497
.
Example (
#8 on p.503
)
Solve the nonlinear system:
3
x
+
y
=
2
x
3
−
2
+
y
=
0
⎧
⎨
⎩
Solution
We can, for example,
(Step 1) Solve the second equation for
y
in terms of
x
and then
(Step 2) Perform a substitution into the first equation.
3
x
+
y
=
2
⇒
3
x
+
2
−
x
3
(
)
=
2
x
3
−
2
+
y
=
0
⇒
y
=
2
−
x
3
Call this
star
.
3
x
+
2
−
x
3
=
2
0
⎧
⎨
⎪
⎩
⎪
3
x
−
x
3
=
0
We may prefer to rewrite this last equation so that the nonzero side
has a positive leading coefficient. We’re more used to that setup.
0
=
x
3
−
3
x
Step 3) Solve
0
=
x
3
−
3
x
for
x
.
Warning: Remember that dividing both sides by
x
is
risky. We may lose solutions. We prefer the Factoring
method.
0
=
x x
2
−
3
(
)
You could factor
x
2
−
3
(
)
over
R
or stop factoring here.

(Sections 7.1-7.3: Systems of Equations)
7.03
Apply the ZFP (Zero Factor Property):
x
=
0
or
x
2
−
3
=
0
x
2
=
3
x
= ±
3
Warning: We’re not done yet! We need to find the corresponding
y
-values.
Step 4) Back-substitute into
star
.
Observe that:
3
(
)
3
=
3
(
)
3
(
)
3
(
)
=
3 3
x
y
=
2
−
x
3
0
2
−
0
( )
3
=
2
3
2
−
3
(
)
3
=
2
−
3 3
−
3
2
− −
3
(
)
3
=
2
− −
3 3
(
)
=
2
+
3 3
Step 5) Write the solution set.
This is usually required if you are solving a system of equations.
Warning: Make sure that your solutions are written in the form
x
,
y
(
)
,
not
y
,
x
(
)
.
The solution set here is:
0,2
(
)
,
3
, 2
−
3 3
(
)
,
−
3
, 2
+
3 3
(
)
{
}
This consists of three real solutions written as ordered pairs.
We assume that ordered pairs are appropriate, since no mention is
made of
z
or other variables.

(Sections 7.1-7.3: Systems of Equations)
7.04
Step 6) Check your solutions in the given system. (Optional)

(Sections 7.1-7.3: Systems of Equations)
7.05
PART C: THE GRAPHICAL METHOD
The Graphical Method for solving a system of equations requires that we graph all of the
equations and then find the resulting intersection points common to
all
the graphs, if any.