When you
solve a system
of equations,
you’re finding
the point where
two lines
intersect.
It’s possible to do this by
actually graphing the lines,
but there are some problems.
It’s time-consuming.
You need graph paper.
You have to graph very neatly.
What if the lines don’t
intersect at whole numbers?
Graphing is usually just about the
least efficient way to solve
systems of equations.
Fortunately there are other ways
to do it.
Often the easiest way to solve
systems of equations is
substitution.
Both in math and in life,
substitution means replacing
one thing with another.
In algebra, substitution means
replacing a variable with
something it is equal to.
Steps for Substitution
1. Solve one of the equations
for one variable in terms of
the other.
(Make it say x = …
or y = …)
2. Substitute into the other
equation.
(Copy the equation,
but change one
variable to what you
just said it equaled.)
3. Solve.
4. Substitute back (to #1) to
find the other answer.
Substitution is usually an
efficient method if at least one
of the variables doesn’t have a
coefficient.
3x + y = 14
2x + 3y = 0
3x + 5y = 2
x – 3y = 10
Use substitution to solve
x + 4y = 11
3x – 2y = 12
x + 4y = 11
3x – 2y = 12
First, solve the top equation
for x in terms of y.
x + 4y = 11
3x – 2y = 12
First, solve the top equation
for x in terms of y.
x = 11 – 4y
x + 4y = 11 … x = 11 – 4y
3x – 2y = 12
Now, substitute for “x” in the
bottom equation.
x + 4y = 11 … x = 11 – 4y
3x – 2y = 12
Now, substitute for “x” in the
bottom equation.
 Change “x” to “11 – 4y”
x + 4y = 11 … x = 11 – 4y
3x – 2y = 12
Now, substitute for “x” in the
bottom equation.
 Change “x” to “11 – 4y”
3(11 – 4y) – 2y = 12
Solve for “y”.
3(11 – 4y) – 2y = 12
Solve for “y”.
3(11 – 4y) – 2y = 12
33 – 12y – 2y = 12
33 – 14y = 12
-14y = -21
y = 1.5
We know …
 y = 1.5
 x = 11 – 4y
Now find x.
We know …
 y = 1.5
 x = 11 – 4y
Now find x.
x = 11 – 4(1.5)
x = 11 – 6
x=5
x + 4y = 11
3x – 2y = 12
The answer is ( 5, 1.5 )
Solve
3x + y = 14
2x + 3y = 0
3x + y = 14
y = 14 – 3x
2x + 3y = 0
2x + 3(14 – 3x) = 0
2x + 42 – 9x = 0
-7x + 42 = 0
-7x = -42
x=6
y = 14 – 3x
x=6
y = 14 – 3(6)
y = 14 – 18
y = -4
So the answer is ( 6, -4 ).
Solve
3x + 5y = 2
x – 3y = 10
3x + 5y = 2
x – 3y = 10
x = 10 + 3y
3(10 + 3y) + 5y = 2
30 + 9y + 5y = 2
14y + 30 = 2
14y = -28
y = -2
x = 10 + 3y
y = -2
x = 10 + 3(-2)
x = 10 – 6
x=4
( 4, -2 )
Solve
y = 2x + 1
y = 6x – 5
Solve
y = 2x + 1
y = 6x – 5
Just change the “y” in one
equation to the expression in
the other.
6x – 5 = 2x + 1
6x – 5 = 2x + 1
4x – 5 = 1
4x
=6
x
= 1.5
6x – 5 = 2x + 1
4x – 5 = 1
4x
=6
x
= 1.5
Since y = 2x + 1
y = 2(1.5) + 1
y=4
… ( 1.5 , 4 )
Here’s a minor variation on
yesterday’s problem …
Two tacos and a burrito cost
$7.02. One taco and two
burritos cost $9.33.
How much is a taco,
and how much is a
burrito?
2t + b = 7.02
t + 2b = 9.33
It’s easy to solve this using
substitution.
2t + b = 7.02
t + 2b = 9.33
We can either get “b” by itself
in the top equation or “t” by
itself in the bottom equation.
2t + b = 7.02
b = 7.02 – 2t
t + 2b = 9.33
t + 2(7.02 – 2t) = 9.33
t + 14.04 – 4t = 9.33
-3t + 14.04 = 9.33
-3t = -4.71
t = 1.57
b = 7.02 – 2t
b = 7.02 – 2(1.57)
b = 7.02 – 3.14
b = 3.88
So … tacos cost $1.57, and
burritos cost $3.88.
If you did it the other way …
2t + b = 7.02
t + 2b = 9.33
t = 9.33 – 2b
2(9.33 – 2b) + b = 7.02
18.66 – 4b + b = 7.02
18.66 – 3b = 7.02
-3b = -11.64
b = 3.88
t = 9.33 – 2b
t = 9.33 – 2(3.88)
t = 1.57
Again … burritos
are $3.88,
and tacos
are $1.57.
Another common type of story
problem is the mixture
problem. Here’s a typical
example …
You have
some
quarters
and some nickels. You have
41 coins that are worth $5.45.
How many of each type of
coin do you have?
In problems like this, write 2
equations:

Number

Value
In problems like this, write 2
equations:


Number
n + q = 41
Value
5n + 25q = 545
or .05n + .25q = 5.45
n + q = 41
5n + 25q = 545
n = 41 – q
n + q = 41
5n + 25q = 545
n = 41 – q
5(41 – q) + 25q = 545
n + q = 41
5n + 25q = 545
n = 41 – q
5(41 – q) + 25q = 545
205 – 5q + 25q = 545
205 + 20q = 545
q = 17
n + q = 41
5n + 25q = 545
n = 41 – q
5(41 – q) + 25q = 545
205 – 5q + 25q = 545
205 + 20q = 545
q = 17
n = 41 – 17 … n = 24
So you have 17 quarters and
24 nickels.
Another method for solving
systems of equations is called
linear combination.
This can also be called the
addition/subtraction method.
The key to linear combination…
Linear combination
(addition/subtraction) works well
when you have terms that are
the same or opposites.
5x + 3y = 7
5x + 7y = 3
4x + 2y = 20
4x – 2y = 4
Steps for Linear
Combination
0. If necessary, multiply one
or both of the equations to
get terms that are the
same or opposites.
(like 3x and -3x … you
usually don’t have to
do this step.)
1. Add or subtract the two
equations to get one of the
variables to cancel out.
2. Solve the easy equation
that’s left.
3. Substitute to get the other
answer.
Solve with linear combination:
5x + 3y = 7
5x + 7y = 3
Solve with linear combination:
5x + 3y = 7
– 5x + 7y = 3
Subtract to cancel the x’s
Solve with linear combination:
5x + 3y = 7
– (5x + 7y = 3)
-4y = 4
Solve with linear combination:
5x + 3y = 7
– (5x + 7y = 3)
-4y = 4
Now solve for y.
Solve with linear combination:
5x + 3y = 7
– (5x + 7y = 3)
-4y = 4
y = -1
Solve with linear combination:
5x + 3y = 7
– 5x + 7y = 3
-4y = 4
y = -1
Substitute back in either
original equation to find x.
Solve with linear combination:
5x + 3y = 7
5x + 7y = 3
5x + 3(-1) = 7
5x – 3 = 7
5x = 10
… x=2
Solve with linear combination:
5x + 3y = 7
5x + 7y = 3
So … the answer is ( 2, -1)
Solve with linear combination:
4x + 2y = 20
4x – 2y = 4
Solve with linear combination:
4x + 2y = 20
4x – 2y = 4
We could actually cancel out
either the x’s or the y’s.
It’s easiest to add the equations
so the y’s cancel.
Solve with linear combination:
4x + 2y = 20
+ ( 4x – 2y = 4)
8x
= 24
x
=3
Solve with linear combination:
4x + 2y = 20
4x – 2y = 4
x
=3
4(3) + 2y = 20
12 + 2y = 20
2y = 8 … y = 4
Solve with linear combination:
4x + 2y = 20
4x – 2y = 4
So … the answer is ( 3, 4 )
Another variation on the taco
problem …
Seven tacos and three burritos
cost $17.34. Two tacos and
three burritos cost
$10.89. How much is a
taco, and how much is
a burrito?
7t + 3b = 17.34
2t + 3b = 10.89
7t + 3b = 17.34
– (2t + 3b = 10.89)
5t
= 6.45
t
= 1.29
7(1.29) + 3b = 17.34
9.03 + 3b = 17.34
3b = 8.31 … b = 2.77
Tacos = $1.29 Burritos = $2.77
Remember Step #0 …
0. If necessary, multiply one
or both of the equations to
get terms that are the
same or opposites.
Here’s when you use it.
4x – 7y = 24
2x + 5y = -5
Nothing is exact the same
or exact opposites.
4x – 7y = 24
2x + 5y = -5 … 4x + 10y = -10
BUT … if you multiply the
bottom by 2, the x’s will be
the same.
4x – 7y = 24
4x + 10y = -10
4x – 7y = 24
– (4x + 10y = -10)
-17y = 34
y = -2
4x – 7y = 24
– (4x + 10y = -10)
-17y = 34
y = -2
4x – 7(-2) = 24
4x + 14 = 24
4x = 10
… x = 2.5
So the answer is ( 2.5 , -2 )