Systems of Equations with No Solution
It is possible for a system of equations to have no solution if there are no values for the variables that would make
all the equations true. For example, the following system of equations has no solution because there are no val-
ues of
x
and
y
that would make both equations true:
3
x
6
y
14
3
x
6
y
9
In other words, one expression cannot equal both 14 and 9.
Practice Question
5
x
3
y
4
15
x
dy
21
What value of
d
would give the system of equations NO solution?
a.
9
b.
3
c.
1
d.
3
e.
9
Answer
e.
The first step in evaluating a system of equations is to write the equations so that the coefficients of one
of the variables are the same. If we multiply 5
x
3
y
4 by 3, we get 15
x
9
y
12. Now we can com-
pare the two equations because the coefficients of the
x
variables are the same:
15
x
9
y
12
15
x
dy
21
The only reason there would be no solution to this system of equations is if the system contains the
same expressions equaling different numbers. Therefore, we must choose the value of
d
that would
make 15
x
dy
identical to 15
x
9
y
. If
d
9, then:
15
x
9
y
12
15
x
9
y
21
Thus, if
d
9, there is no solution. Answer choice
e
is correct.
F u n c t i o n s , D o m a i n , a n d R a n g e
A
function
is a relationship in which one value depends upon another value. Functions are written in the form
beginning with the following symbols:
f
(
x
)
For example, consider the function
f
(
x
)
8
x
2. If you are asked to find
f
(3), you simply substitute the 3
into the given function equation.
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