Systems of Linear
Equations
Math 0099
Section 8.1-8.3
Created and Presented by
Laura Ralston
What is a system of linear
equations?
It is two or more linear equations
with the same variables considered
at the same time.
 Number of variables equals
number of linear equations in the
system
 Examples:

x+y=4
x+ y = -10
x+y=2
3x + 4y = 7
What is the solution set
to a system?


The solution set to the system of linear
equations is ALL ordered pairs that are
solutions to both equations, that is,
makes both equations TRUE at the same
time.
To decide whether an ordered pair is a
solution to a system, substitute the
values for x and y in both equations. If
the results for both equations are true
then the ordered pair is a solution to the
system.
Example: Determine if the given
point is a solution to the system.
x+y=2
3x + 4y = 7
(1, 1)
(4, -2)
Questions to Answer
How do we find the solution, if
there is one?
 Will there always be a solution to a
system of linear equations?
 Can there be more than one
solution?

Methods for Solving a
System of Equations
*
*
*
Graphing - Section 8.1
Substitution - Section 8.2
Addition (Elimination)
Section 8.3
GRAPHING Procedure
1.
2.
3.
4.
Graph the first equation in the
coordinate plane
Graph the second equation on the
same coordinate plane
Record the coordinates of the point
of intersection of the two graphs.
This ordered pair is the solution to
the system
Check solution in both equations.
Three possibilities for
solutions for a system

NO SOLUTION
– Graphically, the
lines would be
parallel.
– Solving for x will
result in a false
statement with no
variable remaining
– INCONSISTENT

ONE SOLUTION
– Graphically, the
lines will intersect
ONCE. Solution
will be an ordered
pair
– Solving for x will
result in a
numerical value
CONSISTENT

INFINITE
SOLUTIONS
– Graphically, the
lines coincide
(same line)
– Solving for x
results in a true
statement with no
variable remaining
– DEPENDENT
Examples


x + 2y = 8
2x – y = 1


y = 2x + 5
4x – 2y = -10
Assignment
Page 595 #1-7 odd, 13-39
odd
SUBSTITUTION



Objective is to eliminate one of the
variables so that a new equation is
formed with just one variable
Most useful when one of the equation is
solved for one variable already OR if
one of the variables has a coefficient of
1; otherwise, you get Fractions !!!
Fractions !!! Fractions !!!
Provides exact answers rather than
estimations
Substitution Steps
1
2
3
4
Solve one of the given equations
for either x or y, whichever is
easier.
Substitute the result from step 1
into the other given equation
Solve for the remaining variable
Substitute (“back substitute”) this
solution into one of the ORIGINAL
given equations
Substitution steps
continued ...
5
6
Solve for the variable. Write final
solution as an ordered pair (x, y)
Check answer in both given
equations. True statements
indicate correct answers.
Examples
x + y =3
 y = 2x

y = 4 – 3x
 Y=-3x + 6

y = -3 =2x
4x – 2y = 6
Assignment
Page 603 #1-41 odd
ADDITION (ELIMINATION)
The idea is to eliminate one of the
variables from the system of linear
equations.
 To do this, one of the variables
must have coefficients that are
opposites.
 Provides exact answers rather than
estimated ones

Addition (Elimination)
Steps
1
2
3
4
Write each equation in standard form
(align like terms)
If needed, multiply one or both
equations by appropriate number(s) so
that the coefficients on either x or y are
opposites.
Add the equations from step 2 together
by combining like terms. This should
result in an equation with one variable.
Solve the equation from step 3.
Addition steps
continued…..
5
6
7
Back Substitute the solution from step 4
into either of the ORIGINAL given
equation
Solve for the other variable. Write final
answer in an ordered pair (x, y)
Check your answer in each original
given equation. True statements result
in correct answers.
Examples




2x + 2y = 4
x – y = -3
y = 3x + 15
6x – 2y = -30


2x – 5y = 6
4x – 10y = -2
Assignment
Page 611 #1-41 odd
COMPASS Practice
Questions
What is the solution of the system of
equations below?
A. (3a, 2a)
B. (-3a, 2a)
3x + 4y = a
C. (15a, 11a)
2x – 4y = 14a
D. (15a, -11a)
E. (3a, -2a)

What are the (x, y) coordinates of
the point of intersection of the
lines determined by the equations
2x – 3y = 4 and y = x?
A. (4, 4)
C. (–4, 4)
E. (2, 0)
B. (–4, –4)
D. (4, –4)