Download Solving Systems of Linear Equations by Graphing

Solving Systems of Linear
Equations by Graphing
Definitions
• A system of linear
equations is two or
more linear equations.
• Ex:
Solution of a system of
linear equations in 2
variables is an ordered
pair of numbers that is a
solution of both
equations in the system.
Example: (0,-4)
How can we find the solution of a
system of linear equations?
• Graphing• Graph each equation
and see where the lines
intersect!
• Graph the system:
• Y = x + 1 and y = 2x - 1
• When we graph we
graph on the same
coordinate system!
• How do we determine if
our graph is correct?
• Substitute the ordered
pair on the graph to
check and make sure it
is a solution
• Y=x+1
• Y = 2x -1
• Example:
3x + 4y = 12
9x + 12y = 36
Solution for the
same line :
Infinite amount of
solutions!
• Example:
3x – y = 6
6x = 2y
Lines that are
parallel do not
have a solution:
Answer: No
solution!
• How can we determine
whether or not we have
a system with infinite
amount of solutions or
no solution?
• Using our slope and y
intercepts!
• To help you find
the solution,
before graphing
write each
equation in
slope intercept
form!
• If the slopes are the
• IF the slopes are the
same and the y
same and the y
intercepts are the same,
intercepts are different,
then you will have an
then you will have
infinite amount of
parallel lines!
solutions!
• If the slopes are
different, then you will
have one solution, an
ordered pair!
Let’s go back and check our examples!
3x + 4y = 12
-3x
-3x
• 9x + 12y = 36
-9x
-9x
4y = -3x + 12
4
4
4
12y = -9x + 36
12
12 12
y = -3x + 3
4
y = -3x + 3
4
3x – y = 6
-3x
-3x
• 6x = 2y
2
2
-y = -3x + 6
-1 -1
Y = 3x or
Y = 3x - 6
y = 3x + 0
Different Types of Systems
• Consistent
Systems: has
at least one
solution
• Inconsistent
Systems: have
no solution
Different Types of Equations
• Independent equations:
Different types of linear
equations (not the
same line)
• Dependent Equations:
the exact same graph
• P. 247
Solving Systems of
Linear Equations
Definitions
• A system of linear
equations is two or
more linear equations.
• Solution of a system of
linear equations in 2
variables is an ordered
pair of numbers that is
a solution of both
equations in the
system.
How can we determine what the
solution is?
•
•
•
•
Guess/Check
Graphing
Substitution
Elimination
Graphing
Guess and Check
• Subsitute all the choices
into BOTH equations!!!!
• 2x – y = 8
• X + 3y = 4
• If the ordered pair is
true for both equations
then it is a system of
the set of linear
equations!
a).
b).
c).
d).
(3, -2)
(-4, 0)
(0, 4)
(4,0)
Example:
-3x + y = -10
X–y=6
a). (-2, 4)
b). (2, 4)
c). (2, -4)
3x + 4y = 12
9x + 12y = 36
a). (0,3)
b). (-4,0)
c). (-4, 6)
• Systems of linear equations can have MORE
THAN ONE SOLUTION!
• These type of systems have an Infinite amount
of solutions!
• Why?
• Y=x–3
• 2y = 2x – 6
• It is the exact same
equation!!!!!!
Let’s try graphing!
*Write the equation in
y = mx + 6
What is the slope?
• Therefore it is the exact
same line and it will
intersect at every single
point!
What is the y intercept?
• 2x – 3y = 6
• -4x + 6y = 5
• Again, let’s write our
equation in y=mx + b
• What is the slope of
each equation and the
y-intercept?
• Try graphing!
• Equations that have the
same slope and
different y-intercepts
are parallel!
• They have NO
SOLUTION!!!!
Summary!
• A system of linear equations can have three
different solutions
– NO solution : the lines are parallel to each (they
have the same slope and different y-intercepts)
– Infinite amount of solutions: The lines are the
same (they have the same slope and same yintercept)
– One solution: Our answer is an ordered pair!