Sat math Essentials

substitution
and 
linear combination
.
Substitution
Substitution involves solving for one variable in terms of another and then substituting that expression into the
second equation.
Example
Here are the two equations with the same solution mentioned above:
2
c
d
11 and 
c
2
d
13
To solve, first choose one of the equations and rewrite it, isolating one variable in terms of the other. It does
not matter which variable you choose.
2
c
d
11 becomes 
d
11 
2
c
Next substitute 11 
2
c
for 
d
in the other equation and solve:
–
A L G E B R A R E V I E W
–
8 8

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c
2
d
13
c
2(11 
2
c
) 
13
c
22 
4
c
13
22 
3
c
13
22 
13 
3
c
9 
3
c
c
3
Now substitute this answer into either original equation for 
c
to find 
d
.
2
c
d
11
2(3) 
d
11
6 
d
11
d
5
Thus,
c
3 and 
d
5.
Linear Combination
Linear combination involves writing one equation over another and then adding or subtracting the like terms so
that one letter is eliminated.
Example
x
7 
3
y
and 
x
5 
6
y
First rewrite each equation in the same form.
x
7 
3
y
becomes 
x
3
y
7
x
5 
6
y
becomes 
x
6
y
5
.
Now subtract the two equations so that the 
x
terms are eliminated, leaving only one variable:
x
3
y
7
(
x
6
y
5)
(
x
x
) 
(
3
y
6
y
)
7 
(
5) 
3
y
12
y
4 is the answer.
Now substitute 4 for 
y
in one of the original equations and solve for 
x
.
x
7 
3
y
x
7 
3(4)
x
7 
12
x
7 
7 
12 
7
x
19
Therefore, the solution to the system of equations is 
y
4 and
x
19.
–
A L G E B R A R E V I E W
–
8 9

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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.