Chapter 3
Section 3.1 – Using Graphs and Tables to Solve Systems
Definition
 A linear system is two or more LINEAR equations in two variables.
An example is
2 x  3 y  6
x  3 y  3
a) The linear equation 2 x  3 y  6 has ______________ solutions. Find 3 solutions and GRAPH
b) The linear equation x  3 y  3 has ______________ solutions. Find 3 solutions and GRAPH
1
c) Do you think that the equation 2 x  3 y  6 and x  3 y  3 have a COMMON solution? If so
what is it and how do you know?
d) Is  3,0 a solution to BOTH equations? CHECK!!
e) GRAPH both equations on the same grid.
f) Do the lines INTERSECT? At what point do they intersect?
Activity #11
2
Continued Definitions….
 A linear system has a SOLUTION when …….
 A linear system has NO solution when ……..
 A linear system has infinitely many solutions when……
 A linear system with ONE solution is called a CONSISTENT System.
 A linear system with NO solution is called an INCONSISTENT System.
 A linear system with MANY solutions is called a DEPENDENT System.
Example 1
Find the solution set of the system by graphing.
2 x  3 y  6
x  3 y  3
3
Example 2
Find the solution set of the system by graphing.
3 y  2  23x  1  7
y  7  4x  2
Example 3
Find the solution set of the system by graphing.
y  2x  1
y  2x  5
4
Section 3.2 – Substitution Method and Elimination Method.
We don’t want to graph lines ALL the time to solve linear systems, so by using the substitution method
and the elimination method, we will solve linear systems without graphing each equation.
Solve the linear system
2 x  3 y  6
x  3 y  3
STEPS for the SUBSTITUTION METHOD
1. Rewrite both equations in SLOPE INTERCEPT form
2. Set Equations equal to each other. (YOU SHOULD HAVE NO y variables and only ONE equation)
3. Solve for x
4. Substitute the value you found for x into one of the ORIGINAL equations. It doesn’t matter
which one you pick….!!!
5. Simplify, and you should have the y value.
6. Final answer should be in ordered pair notation.
5
Example 4
Using the substitution method, solve the following systems.
a)
4 x  3 y  14
y  2x  8
b)
yx
y  x
1
x5
c)
2
2 x  3 y  1
y
6
Solve the following linear system by using the elimination method.
2 x  3 y  6
x  3 y  3
STEPS for the ELIMINATION METHOD
1. Line up your variables and constants
2. Pick a variable to ELIMINATE
3. Multiply the TOP or BOTTOM or BOTH equations by a constant to make OPPOSITES
4. Combine your equations together. (You should have ONE equation and ONE variable)
5. Solve for the remaining variable
6. Go back to ONE of the original equations and substitute the value you found on STEP 5
7. Final answer should be in ordered pair notation.
7
Example 5
Solve the following linear systems using the elimination method.
a)
6 x  5 y  13
9x  4 y  2
b)
2x  3  3 y  1  5
 4x  2  5 y  3  13
8
Example 6
Solve the linear systems using either method. (substitution, elimination, or graphing).
Please state the method you are using.
Please state what system it is
a)
4x  3y  1
 20 x  15 y  3
b)
4x  5 y  3
 12 x  15 y  3
9
Example 7
Solve the following linear equation.
2 x  y 3x  y

1
3
6
a)
x  y 2 x  y 31


3
4
32
b) Find all SOLUTIONS!!
Find the linear system
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
-1
2
3
4
5
x
-2
-3
-4
-5
10
Section 3.3 – Using Systems to Model data
Activity 12
Example 1
The number (in millions) of US households with PCs is C t  and the total number (in millions) of US
household H t  are modeled by the system, where t is the number of years since 2000.
C t   3.18t  67.92
H t   1.19t  107.36
a) Find the slopes of the C and H. What do they mean in this situation?
b) Predict the percentage of US households that will have PCs in 2013
c) Use substitution or elimination to predict when all US households will have PCs . Check your work
using your graphing calculator.
11
Example 2
The percentage of Continental Airlines tickets that were free and the percentages of American Airlines
tickets that were free are show in the table below for various years.
Year
1999
2000
2001
2002
2003
2004
2005
Percent of Tickets that
were Free
Continental American
6.0
10.3
6.4
9.8
6.2
9.5
6.6
9.6
7.3
9.2
7.3
8.7
7.4
8.6
a) Let c t  be the percentage of Continental Tickets that are free and a t  be the
percentage of American tickets that are free, both at t years since 1990. Find equations
of c and a .
b) Predict when the percentage of Continental tickets that are free will be equal to the
percentage of American tickets that are free. What will be the percentage?
c) Use “interest” on a graphing calculator to verify your work.
12
Section 3.4 – Word Problems
Problem 2 in your book
A 9000-seat amphitheater has tickets for sale at $35 and $62. How many tickets should be sold at each
price for a sellout performance to generate a total revenue of $382,500?
SOLUTION:
 What are the unknowns?
 Is there a relationship between these variable?
$35 Tickets
$62 Tickets
Total Tickets Sold
9000
LINEAR EQUATION THAT DESCRIBES THE ABOVE TABLE IS ______________________
Money made with
$35 Tickets
Money made with
$62 Tickets
Money in Total
382,500
LINEAR EQUATION THAT DESCRIBES THE ABOVE TABLE IS ________________________
Since we are trying to do BOTH sell 9000 tickets AND make $382,500, then both linear equations
MUST be satisfied…. In other words we are looking for A SOLUTION TO BOTH EQUATIONS!!
a) SET UP A SYSTEM and SOLVE IT!!!
b) Answer the question in a complete sentence.
13
Problem 8 in book.
A new college may open in the future. Full-time instructors would teach 15 units each semester.
Researchers estimate that part-time instructors teach an average of 6 units each semester and that
there is student demand for a total of 3300 units of courses. There would be enough office space for
250 instructors. To keep costs low by hiring as many adjunct instructors as possible, yet have office
space for all instructors, how many full-time instructors and adjunct instructors shoe be hired?
Problem 16 in book
A person plans to invest $9000. He will invest in both a Winslow Green Growth account at 17% annual
interest and a Columbia Small Cap Growth I Z account at 10% annual interest. How much should he invest
in each account so that the total interest in 1 year will be $1040?
14
Section 3.5 – Using Linear Inequalities in One Variable to Make Predictions
Inequalities are very similar to equations. Which one is an equation? How do you know?
3x  4  5 x  3
3x  4  5 x  3
Inequalities Symbols – Using complete sentences, describe the following inequalities.
 x  3 ___________________________________________________________
 x  5 ___________________________________________________________
 x  1____________________________________________________________
 x  10 ____________________________________________________________
Graphing Inequalities
a) Graph the solution x  
3
7
b) Graph the solution x  4 and write an interval notation.
c) Graph the solution x  4 and write an interval notation.
15
d) Graph the solution x 
3
and write an interval notation.
5
e) Graph the solution x  
13
and write an interval notation.
4
Solving Inequalities
Solving an inequality is very similar to solving an equation.
1. Solve the following equation and STATE THE PROPERTIES you are using.
3x  5  4
2. Solve the inequality and STATE the PROPERTIES you are using.
3x  5  4
16
Properties of Inequalities
Addition Property of Inequality
Subtraction Property of Inequality
Multiplication Property of Inequality
Division Property of Inequality
3. Solve the following inequality and STATE the Properties you are using. Write your solution using
interval notation and GRAPH your solution set.
2x  3  8
17
Example 4
Solve the following. Please STATE the PROPERTIES you are using. WRITE your solution using interval
notation and GRAPH your solution set.
a)  5x  15  2 x
b) 6.6  5.2 x  20.64
c) 13.5  4.1w  2w  6.4  5.9w
18
d) 
3
5 7 5
x   x
4
2 8 2
e)  5  3x  1  13
f) 7  5  2 x  13
Activity 16
19