Systems of Linear Equatiosn
A system of linear equations is two or more
linear equations that are being solved
simultaneously.
In general, a solution of a system in two
variables is an ordered pair that makes BOTH
equations true.
Solutions of a System
The solution of a system is where two graphs
intersect or what they have in common. If an
ordered pair is a solution to one equation, but
not the other, then it is NOT a solution to the
system.
A consistent system is a system that has at
least one solution.
An inconsistent system is a system that has no
solution.
Solution of a System
The equation of a system is dependent if ALL
the solutions of one equation are also
solutions of the other equation. In other
words, they end up being the same line.
The equations of a system are independent if
they do not share ALL solutions. In other
words, they have one point in common.
Solution of a System
There are three possible outcomes:
 One solution
 No solution
 Infinitely many solutions
One Solution
If the system in two variables has one
solution, it is an ordered pair that is a solution
to BOTH equations.
The graph here illustrates a system of two
equations and two unknowns that has one
solution:
No Solution
If the two lines are parallel to each other, they will
never intersect. This means they do not have any
points in common. In this system, you would have
no solution.
This graph illustrates a system of two equations and
two unknowns that has no solutions.
Infinite Solutions
If two lines end up lying on top of each other, then
there is an infinite number of solutions. They end up
being the same line, so any solution that would work
in one equation is going to work in the other.
This graph illustrates a system of two equations and
two unknowns that has an infinite number of
solutions.
Practice
Determine whether each ordered pair is a solution of the system.
(3, -1) and (0, 2)
Solve by Graphing
Solve by Graphing
Solve by Graphing
Substitution Method
To solve a system using substitution, solve one
equation for either x or y.
Then substitute this equation into the other
equation for the variable you solved for in the
first.
Practice
3y - 2x = 11
y + 2x = 9
Practice
−5x + y = −3
3x − 8y = 24
Practice
−2x + 6y = 6
−7x + 8y = −5
Elimination Method
First, be sure that the variables are "lined up" under one
another. In this problem, they are already "lined up“
x - 2y = 14
x + 3y = 9
Decide which variable ("x" or "y") will be easier to
eliminate. In order to eliminate a variable, the numbers
in front of them (the coefficients) must be the same or
negatives of one another. Looks like "x" is the easier
variable to eliminate in this problem since the x's already
have the same coefficients.
Elimination Method
Now, in this problem we need to subtract to eliminate
the "x" variable. Subtract ALL of the sets of lined up
terms.
x - 2y = 14
-x - 3y = - 9
- 5y = 5
Solve this simple equation.
-5y = 5
y = -1
Elimination Method
Substitute "y = -1" into either of the ORIGINAL
equations to get the value for "x".
x - 2y = 14
x - 2(-1) = 14
x + 2 = 14
x = 12
Check: substitute x = 12 and y = -1 into BOTH
ORIGINAL equations. If these answers are correct,
BOTH equations will be TRUE!
x - 2y = 14
x + 3y = 9
12 + 3(-1) = 9
12 - 2(-1) = 14
12 - 3 = 9
12 + 2 = 14
9 = 9 (check!)
14 = 14 (check!)
More Challenging
You can probably see the dilemma with this problem
right away. Neither of the variables have the same (or
negative) coefficients to eliminate.
4x + 3y = -1
5x + 4y = 1
Challenge!
In this type of situation, we must MAKE the coefficients the
same (or negatives) by multiplication. You can MAKE
either the "x" or the "y" coefficients the same. In this
problem, the "y" variables will be changed to the same
coefficient by multiplying the top equation by 4 and the
bottom equation by 3.
Remember:
* you can multiply the two differing coefficients to obtain
the new coefficient if you cannot think of another smaller
value that will work.
* multiply EVERY element in each equation by your
adjustment numbers.
4(4x + 3y = -1)
16x + 12y = -4
3(5x + 4y = 1)
15x + 12y = 3
No longer Challenging!
Now, in this problem we need to subtract to eliminate the
"y" variable.
16x + 12y = -4
-15x - 12y = - 3
x
=-7
Substitute "x = -7" into either of the ORIGINAL equations to
get the value for "y".
5x + 4y = 1
5(-7) + 4y = 1
-35 + 4y = 1
4y = 36
y=9
Elimination Method
Check: substitute x = -7 and y = 9 into BOTH
ORIGINAL equations. If these answers are correct,
BOTH equations will be TRUE!
 4x + 3y = -1
4(-7) +3(9) = -1
-28 + 27 = -1
-1 = -1 (check!)
 5x + 4y = 1
5(-7) + 4(9) = 1
-35 + 36 = 1
1 = 1 (check!)
Practice
4x + 8y = 20
−4x + 2y = −30
Practice
8x + y = −16
−3x + y = −5
Practice
−3x + 7y = −16
−9x + 5y = 16
Practice
5x + 4y = −30
3x − 9y = −18