Solving Systems of Linear Equations – What Should You Do??
GRAPHING
1.) To solve a
system of
equations by
graphing the
equations, start by
graphing the first
equation in a
coordinate plane.
2.) Graph the second
equation in the
same coordinate
plane.
3.) Find the point
where the two
lines intersect.
Write the point as
an ordered pair in
the form (x,y). The
ordered pair
indicates the x and
y value that solves
both equations.
CHECK YOUR SOLUTION!
EXAMPLE
4x  y  5
2x  3y  15
4x  y  5
y  4 x  5
b5
m  4
2x  3y  15
3y  2x  15
2
y
x5
3
b  5
2
m
3
When to Use/Be Careful
When to use:
1.) To have a visual
representation of
the system of
linear equations.
2.) When solving
algebraically may
be difficult, using
a graphing
calculator gives
quick, accurate
answer,
i.e.:
324 p  456t  225
178 p  245t  150
Be Careful:
1.) To find two
correct points on
each line.
2.) To graph carefully
so you read the
correct point of
intersection.
1
Solving Systems of Linear Equations – What Should You Do??
SUSTITUTION
Example
4x  y  5
1.) Choose one equation
and solve for one
2x  3y  15
variable in terms of the
other variable.
2.) In the other equation,
substitute the value of
the variable you solved
for.
3.) Distribute, combine like
terms, and solve the
equation for the one
variable.
4.) Place the value of the
variable you
determined in step 3
into the equation you
defined in step1 to
solve for the other
variable.
5.) Write the point as an
ordered pair in the form
(x,y). The ordered pair
indicates the x and y
value that solves both
equations.
CHECK YOUR SOLUTION!
1.) For this example pick
Equation #1 and solve for
y:
4x  y  5
y  5  4x
2.) Now substitute this value
of y in Equation #2:
2x  3y  15
2x  3(5  4x)  15
3.) Distribute the 3, and
combine like terms, solve
for x
2x  15  12x  15
10x  15  15
10x  30
x3
4.) Now find the value of y:
y  5  4x
y  5  4(3)
y  5  12
y  7
5.)
When to Use/Be Careful
When to use:
1.) When an equation
in your system is
solved for 1
variable (i.e.
y=2x)
2.) When an equation
in your system
has a coefficient
of 1 or -1
3.) When it is easy to
solve for one
variable in terms
of the other
Be Careful:
1.) To completely
solve for one
variable in terms
of the other
(solved variable
should have a
coefficient of +1)
2.) Distribute
correctly
(3,-7)
2
Solving Systems of Linear Equations – What Should You Do??
ELIMINATION
Example
1.) Evaluate the equations in
4x  y  5
your system and
determine if there are
2x  3y  15
terms that are opposites.
2.)
3.)
4.)
5.)
If your two equations do
not have opposite terms
multiply one or both
equations by the
appropriate constant so
you do have opposite
terms.
Use the Addition
Property of Equality to
add the equations
together, the result will
be 1 equation with 1
unknown.
Solve the equation
Plug-in the value for the
variable solved in step 3
into EITHER original
equation to find the
value of the second
variable.
Write the point as an
ordered pair in the form
(x,y). The ordered pair
indicates the x and y
value that solves both
equations.
Solution
1.) There are no opposite terms,
so multiply Equation 1 by (-3)
to get opposite terms:
3(4x  y  5) 12x  3y  15

2x  3y  15
2x  3y  15
2.) Add the equations:
10x  30
3.) Solve for x
x3
4.) Solve for y
4x  y  5
4(3)  y  5
12  y  5
y  7
5.) (3,-7)
When to Use/Be Careful
When to use:
1.) If your system of
equations has
opposite terms
2.) If you can easily
multiply one or
both equations in
your system to
have opposite
terms.
3.) If graphing and
substitution are not
easy methods.
Be Careful:
1.) To have opposite
terms before
adding your
equations
2.) When multiplying
an equation by a
constant multiply
the ENTIRE
equation by the
constant.
CHECK YOUR SOLUTION!
3
Solving Systems of Linear Equations – What Should You Do??
4