Learning Objectives for Section 4.1

1
Learning Objectives for Section 4.1
Review Systems of Linear Equations in Two
Variables

• After this lesson, you should be able to
• solve systems of linear equations in two
  variables by graphing
• solve these systems using substitution
• solve these systems using elimination by addition
• solve applications of linear systems.

2
System of Linear Equations
System of Linear Equations refers to more than
one equation being graphed on the same set of
coordinate axes.
Solution(s) of a System where the lines
intersect. Also, the solutions coordinates will
work in all of the systems equations.
3
Solving Graphically

1. On a single set of coordinate axes, graph each
   equation and label them.
2. Find the coordinates of the point where the
   graphs intersect. These coordinates give the
   solution of the system. Label this point.
3. If the graphs have no point in common, the system
   has no solution.
4. Check the solution in both of the original
   equations.

4
Solve by Graphing

• Example 1 Solve the following system by
  graphing

y
x
Check your solution!
5
Checking
Solution
6
Consistent Systems
Consistent system- a system of equations that
has a solution.
(the system has at least one point of
intersection)
one solution exists
infinitely many solutions exist
7
Inconsistent Systems
Inconsistent system- a system of equations that
has no solutions.
(the lines are parallel)
no solution exists
8
Independent and Dependent Equations
Independent Equations- the equations graph
different lines
one solution exists
no solution exists
Dependent Equations- the equations graph the same
line
infinitely many solutions exist
9
Solving a System on the Calculator
Step 1 Graph each equation. (Use your
calculator.) 4x y 9 x 3y 16
Make sure the equations are in slope-intercept
form!
Step 2 Find the coordinates of the point of
intersection.
2nd
CALC
5 intersect
First curve? (make sure cursor is on the line
for y1.)
Second curve? (make sure cursor on the line for
y2.)
Guess? (cursor to where you feel the
intersection is)
ENTER
ENTER
ENTER
10
Calculator Example Continued
Solution
What type of system is this?
What type of equations are these?
11
Example
Example Solve the system by graphing on your
calculator.
12
Example with Dependent Equations
Example Solve the system by graphing on your
calculator.
13
Solving Method of Substitution

1. Solve one of the equations for either x or y.
2. Substitute that result into the other equation to
   obtain an equation in a single variable (either x
   or y).
3. Solve the equation for that variable.
4. Substitute this value into any convenient
   equation to obtain the value of the remaining
   variable.
5. Check solution in BOTH ORIGINAL equations.
6. Write your solution as an ordered pair. If there
   is no solution, state that the system is
   inconsistent.

14
Example

• Example Solve the system using substitution.

15
Another Example

• Example Solve the system using substitution.

16
Solving Method of Elimination by Addition Method

1. Write both equations in the general form Ax By
   C.
2. Multiply the terms of one or both of the
   equations by nonzero constants to make the
   coefficients of one variable differ only in sign.
   
3. Add the equations and solve for the variable.
4. Substitute the value into one of the ORIGINAL
   equations to find the value of the other
   variable.
5. Check the solution in BOTH ORIGINAL equations.
6. Write your solution as an ordered pair. If there
   is no solution, state that the system is
   inconsistent.

17
Addition Method Example
Example Solve the system using addition
method.
18
Another Addition Method Example
Example Solve the system using addition
method.
19
Another Example
Example Solve the system using addition
method.
20
Examples
Example Solve the system using ANY method.
21
Examples
Example Solve the system using ANY method.
22
Special Cases

• When solving a system of two linear equations in
  two variables
• If an identity is obtained, such as 0 0, then
  the system has an infinite of solutions.
• The equations are dependent
• The system is consistent.
• If a contradiction is obtained, such as 0 7,
  then the system has no solution.
• The system is inconsistent.
• The equations are independent.

23
Application

• A man walks at a rate of 3 miles per hour and
  jogs at a rate of 5 miles per hour. He walks and
  jogs a total distance of 3.5 miles in 0.9 hours.
  How long does the man jog?

24
Application

• A man walks at a rate of 3 miles per hour and
  jogs at a rate of 5 miles per hour. He walks and
  jogs a total distance of 3.5 miles in 0.9 hours.
  How long does the man jog?
• Solution
• Let x represent the amount of time spent walking
• Let y represent the amount of time spent jogging.

25
Supply and Demand
The quantity of a product that people are willing
to buy during some period of time depends on its
price. Generally, the higher the price, the less
the demand the lower the price, the greater the
demand. Similarly, the quantity of a product
that a supplier is willing to sell during some
period of time also depends on the price.
Generally, a supplier will be willing to supply
more of a product at higher prices and less of a
product at lower prices. The simplest supply and
demand model is a linear model where the graphs
of a demand equation and a supply equation are
straight lines.
26
Supply and Demand(continued)
In supply and demand problems we are usually
interested in finding the price at which supply
will equal demand. This is called the
equilibrium price, and the quantity sold at that
price is called the equilibrium quantity. If we
graph the the supply equation and the demand
equation on the same axis, the point where the
two lines intersect is called the equilibrium
point. Its horizontal coordinate is the value of
the equilibrium quantity, and its vertical
coordinate is the value of the equilibrium price.
27
Supply and DemandExample
Example Suppose that the supply equation for
long-life light bulbs is given by p 1.04 q -
7.03, and that the demand equation for the bulbs
is p -0.81q 7.5 where q is in thousands
of cases. Find the equilibrium price and
quantity, and graph the two equations in the same
coordinate system.
28
Supply and Demand(Example continued)
If we graph the two equations on a graphing
calculator and find the intersection point, we
see the graph below.
Demand Curve
Supply Curve
Thus the equilibrium point is (7.854, 1.14), the
equilibrium price is 1.14 per bulb, and the
equilibrium quantity is 7,854 cases.
29
Now, Another Example

• A restaurant serves two types of fish dinners-
  small for 5.99 each and large for 8.99. One
  day, there were 134 total orders of fish, and the
  total receipts for these 134 orders was 1024.66.
  How many small dinners and how many large dinners
  were ordered?

30
Solution

• A restaurant serves two types of fish dinners-
  small for 5.99 each and large for 8.99. One
  day, there were 134 total orders of fish, and the
  total receipts for these 134 orders was 1024.66.
  How many small dinners and how many large dinners
  were ordered?