3.1
Solve Linear Systems by Graphing
System of two linear equations
Two
equations, with the variables x and y that can be written as:
Ax + By = C Equation
1
Dx + Ey
= F Equation 2
Solution of a system
An ordered pair (x, y) that satisfies
each equation
Consistent
A system that has at least one solution
Inconsistent
A system that has no solution
Independent
A consistent system that has exactly one
solution
Dependent
A consistent system that has infinitely many
solutions
Example
1
Solve
a system graphically
Graph
the system and estimate the solution. Then check the solution algebraically.
4x + 2y = 4 Equation
1
2x - 3y = 10 Equation 2
Solution
Graph both equations. The lines appear to intersect at (_2_, _-2_). Check
this algebraically as follows:
|
Equation
1 |
Equation
2 |
|
4x + 2y = 4 |
2x - 3y
= 10 |
|
4(_2_)
+ 2(_-2_) = ? =4 |
2(_2_)
-3(_-2_) =?
= 10 |
|
_4_ = 4
ü |
_10_ =
10 ü |
Remember
to check the visual solution in both equations.
Graph the linear system and estimate the solution.
Then check the solution algebraically.
1.
4x + y = -2
-6x - 3y = 12
(1, -6)
NUMBER
OF SOLUTIONS OF A LINEAR SYSTEM
|
Exactly one solution |
Infinitely many solutions |
No Solutions |
|
|
|
|
|
Lines intersect at one point consistent
and independent Infinitely many solutions it y |
Lines coincide; consistent and
dependent |
Lines
are _parallel_; _inconsistent_ |
Example
2
Solve
a system with many solutions
Solve
the system.
Then classify the system as consistent and independent, consistent and
dependent, or inconsistent
-2x + y = 4 Equation 1
4x
- 2y = -8 Equation
2
The graphs of the
equations are _the same line_. So, each point on the line is a solution,
and the system has _infinitely many_ solutions. Therefore, the system is
_consistent and dependent_.
Example
3
Solve
a system with no solution
Solve
the system. Then
classify the system as consistent and independent, consistent and dependent,
or inconsistent.
-2x
+ 4y = 8 Equation 1
-2x
+ 4y = -4 Equation 2
Solution
The
graphs of the equations are two _parallel lines_. Therefore, the system
is _inconsistent_.
Example
4
Writing
and using a linear system
Ice
Cream
Shop At an ice cream shop, one customer pays $9 for 2 sundaes and 2 milkshakes.
A second customer pays $13 for 2 sundaes and 4 milkshakes. How much do each sundae and milkshake cost?
Total
cost (dollars) Cost
per sundae (dollars/ shake) Cost
per shake (dollars/ shake) Number
of shakes Number of sundaes
Verbal
model
= · + ·
_9_ = _2_ · x+_2_ ·y Equation 1 (Customer 1)
_13_ = _2_ · x + _4_ ·y Equation 2
(Customer 2)
Graph
the equations
_2_
x + _2_ y = _9_ and
_2_ x + _4_
y = _13_.
The
lines appear to intersect at about the point (_2.5_, _2_).
Check
this algebraically.
_2_(_2.5_)
+ _2_(_2_) = _5_ + _4_ = 9 ü Equation
1 checks.
_2_(_2.5_)
+ _4_(_2_) = _5_ + _8_ = 13 ü Equation
2 checks.
The
solution is (2.5, _2_). So, each sundae costs $ 2.50 and
each milkshake costs $ 2.00.