SYSTEMS OF EQUATIONS
Unit 2
A system of equations is two or
Systems
of with
Equations
more
equations
the same
variables.
ex. 3x-2y=8
4x+3y=12
There are several ways to solve
systems.
1. solve by graphing
2. solve by substitution
3. solve by linear combination
4. solve using matrices
Solving by Graphing
• When solving a system by graphing you
need to graph each line and look for a
point of intersection.
• The point (x,y) is the solution to the
system.
• To be a solution, it must satisfy both
equations.
Classifying Systems
• If a system has at least one solution, then
it is called consistent
• If a system does not have a solution then it
is called inconsistent
• If the system is consistent and has one
solution, then it is called independent
• If the system is consistent and has an
infinite number of solutions, then it is
called dependent.
Solving by Graphing
• If the lines intersect at one point, then
there is one solution
• If the lines do not intersect at all, then
there is no solution
• If the lines intersect at every point, then
there is an infinite number of solutions
Examples
x  y  4
3 x  4 y  12
3x  y  8
4 y  12  3 x
Examples
3x  y  2
 3x   y  1
Solving Algebraically
•
1.
Substitution Method—
Solve one equation for one variable (best to
solve for variable with a coefficient of 1 if
possible)
2. Substitute for the variable you just solved
for in first step into the other equation
3. Once you find the value of one variable,
substitute into either equation to find value
of other variable
x  2y  8
1
x  y  18
2
Example
Example
1
x y 5
2
3y  2x  6
Unique Solutions
• When solving by substitution method
and the variables cancel out and the
statement you are left with is false, like
5=8, then there is no solution to the
system
• When solving and the variables cancel
out and the statement you are left with
is true, then there is an infinite # of
Solutions (not all reals!)
Examples
9 x  6 y  24
3 x  4 y  12
6 x  4 y  16
6 x  8 y  16
Linear Combination
also called addition method or
elimination method
• Add the two equations together so that
one variable cancels out (may need to
multiply one or both of the equations by
some factor that would cause a variable to
cancel
• Need to have equation in standard form
Examples
4 x  2 y  15
2x  2 y  7
More examples
3 x  7 y  14
5 x  2 y  45
More Examples
y  2x 1
 4 x  2 y  2
Real World Examples
• Assign variables
• Set up equations
• Solve
• Ex. The perimeter of a rectangle is 36
inches. The length is twice the width.
Find the width.
Real World Examples
• Ex: Thirty people are going to lunch. It
costs $5 for kids and $12 for adults. The
total bill without tax was $220. How many
kids and adults went to lunch?
Systems with 3 equations and 3
variables
Steps:
1. Take 2 equations and eliminate a variable
2. Take two different equations and eliminate
the SAME variable
3. Take the 2 new 2 variable equations and
eliminate one of those variables
4. Plug the value into either 2 variable eq. for
the value of the 2nd variable
5. Plug both values into original eq. to find
value of 3rd variable
Example
x  2 y  z  10
2 x  y  3 z  5
2 x  3 y  5 z  27
Work Space
• (7,4,-5)
More Examples
2x  3y  z  0
x  2 y  4 z  14
3 x  y  8 z  17
Work Space
(4,-3,-1)
One more Example—don’t
complain!!!
3x  y  z  5
3 x  2 y  z  11
6 x  3 y  2 z  12
Work Space
• (2/3,2,-5)
Try one more…
3x  5 y  3
10 y  2 z  2
x  z
• (-2,3/5,2)
Homework
 4a  8
5a  2c  0
7b  3c  22
4a  2b  6c  2
6a  3b  9c  3
8a  4b  12c  6
4 x  2 y  8 z  30
x  2 y  7 z  12
2 x  y  4 z  15
Hw answers
1. (2,1,5)
2. (1,5,7)
3. ( no sol )
4. (inf # sol )
System of Linear Inequalities
• To graph a linear inequality, you graph the
related linear equation and shade the area
that contains the solution
• < or > symbols have a boundary line that
is dotted—if it is ≤ or ≥ then the boundary
line is solid
• Shade the region that includes x and y
values that make the inequality true (test a
point)
System of Linear Inequalities,
cont.
• 2 or more inequalities
• Graph each inequality and shade
• Darken where the two shaded regions
overlap—this will be the solution to the
system
Example
yx
y  x 1
2 y
Example
x0
y0
y  2x  5
Example
0 y4
1  x  5
x y
Linear Programming
•
A method used to find optimal solutions such
as maximum or minimum profits
Steps:
1. Assign variables
2. Determine the constraints (inequalities)
3. Find the feasible region (area of solution)
4. Determine the vertices of feasible region
5. Plug those values into the profit equation(also
called objective function)
Example
• Mr. Farmer wants to plant some corn and
wheat and he gets the following statistics
from the US Bureau of Census
Crop
Corn
Yield per acre
113.5 bu
Avg Price
$3.15/bu
Soybeans
34.9 bu
$6.80 bu
Wheat
35.8 bu
$4.45 bu
Cotton
540 lb
$0.759/lb
Rice, rough
5621 lb
$0.0865/lb
Example continued
• Mr. Farmer can have no more 120 acres
of corn and wheat
• At least 20 and no more than 80 acres of
corn
• At least 30 acres of wheat
How many acres of each crop should Mr.
Farmer plant to maximize the revenue
from his harvest?
Working through the problem..
• Assign Variables
X=acres of corn and
y=acres of wheat
• List the constraints
20  x  80
y  30
x  y  120
Graph it
Example continued
• List the vertices
• Determine Profit equation
• Which would yield the most?
Another Ex.
• A snack bar cooks and sells hamburgers and hot dogs
during football games. To stay in business, it must sell
at least 10 hamburgers but cannot cook more than 40. It
must also sell at least 30 hot dogs but cannot cook more
than 70. The snack bar cannot cook more than 90 items
total. The profit on a hamburger is $0.33 and on a hot
dog it is $0.21. How many of each item should it sell to
make the maximum profit?
• Profit Equation: __________________________
• Constraints:
• Answer: _________________________
Another Example
• As a receptionist for a veterinarian, Sue
scheduled appointments. She allots 20 minutes
for a routine office visit and 40 minutes for
surgery. The vet can not do more than 6
surgeries per day. The office has 7 hours
available for appointments. If an office visits
costs $55 and most surgeries costs $125, find a
combination of office visits and surgeries that will
maximize the income the veterinarian practice
receives per day.
What do you know…
• Assign variables
x=number of office visits
y=number of surgeries
• Constraints: 7 hours needs to be in terms of
minutes
x0
y0
20 x  40 y  420
y6
Continued…
• Graph and determine
coordinates of the
vertices
• You should get (0,0)
(0,6)(9,6)(21,0)
Continued….
• Determine the profit equation
• $55v + $125s = P
• Test the points
• Highest profit would be when there are 6
surgeries and 9 visits