Transcript 4.1 Systems of Linear Equations in two variables

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.
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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 solution’s coordinates will work in all of the
system’s equations.
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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.
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Solve by Graphing
Example 1: Solve the following system by graphing:
y
x  y  4

2 x  y  5
x
Check your solution!
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Checking…
x  y  4

2 x  y  5
Solution:
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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
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Inconsistent Systems
Inconsistent system- a system of equations that has no solutions.
(the lines are parallel)
no solution exists
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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
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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?
Second curve?
(make sure cursor is
(make sure cursor on
the line for y2.)
on the line for y1.)
ENTER
ENTER
Guess?
(cursor to
where you feel the
intersection is)
ENTER
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Calculator Example Continued…
4 x  y  9

 x  3 y  16
Solution:
What type of system is
this?
What type of equations
are these?
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Example
Example: Solve the system by graphing on your calculator.
3x  5  2 y

3x  2 y  7
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Example with Dependent Equations
Example: Solve the system by graphing on your calculator.
5
 2 x  3 y  6

y   5 x  2

6
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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
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that the system is inconsistent.
Example
Example: Solve the system using substitution.
x  3y  9

2 x  y  10
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Another Example
Example: Solve the system using substitution.
3x  2 y  7

 y  2x  3
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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.
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Addition Method Example
Example: Solve the system using addition method.
2 y  3x

2 x  7  y
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Another Addition Method Example
Example: Solve the system using addition method.
5 x  16  7 y

2 x  8 y  26
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Another Example
Example: Solve the system using addition method.
2 x  5 y  6

4 x  10 y  1
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Examples
Example: Solve the system using ANY method.
x  2 y  8

x  4  2 y
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Examples
Example: Solve the system using ANY method.
2 x  13 y  120

14 x  91y  840
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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.
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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?
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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.
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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.
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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.
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Supply and Demand
Example
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.
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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.
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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?
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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?
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