Download Chapter 6 Notes

Algebra Chapter 6 Notes
Systems of Equations and Inequalities
Lesson 6.1 Solve Linear Systems by Graphing
System of linear equations: _______________________________________________________
______________________________________________________________________________
Solution of a system of linear equations: _____________________________________________
____________________________________________________________________________________
Consistent independent system: ___________________________________________________________
_____________________________________________________________________________________
Example 1: Use the graph to solve the system. Then check your solution algebraically.
2x + y = 4
3x – 5y = 6
Equation 1
Equation 2
Example 2: Solve the linear systems by graphing.
y = 3x – 2
y = -2x + 3
Equation 1
Equation 2
x – 3y = 2
–5x + y = 4
Equation 1
Equation 2
Example 3: The Rosebud Flower Shop has a basic delivery charge of $5 plus a rate of $.25 per
mile. The Beautiful Bouquets Shop has a basic delivery charge of $7 plus a rate of $.20 per mile.
Determine the number of miles a delivery must be for the charges to be equal.
Lesson 6.2 Solve Linear Systems by Substitution
Example 1: Solve the linear system using substitution.
y = 2x – 3
Equation 1
x + 3y = 5
Equation 2
Example 2: Solve the linear system using substitution.
2x + 5y = 5 Equation 1
x – 4y = 9
Equation 2
Example 3: A food cooperative is a business that usually offers special prices on locally grown
food and produce. Some cooperative are clubs and other are retail stores. The weekly costs for
seasonal produce offered by a club-based and a store-based food cooperative are shown in the
table. Find the number of weeks at which the total cost of weekly produce will be the same.
Type of cooperative
Club
Retail
Club fee
$20
None
Cost per week
$15
$17.50
Example 4: A chemist needs 15 liters of a 60% alcohol solution. The chemist has a solution that
is 50% alcohol. How many liters of the 50% alcohol solution and pure alcohol should the
chemist mix together to make 15 liters of 60% alcohol solution?
Lesson 6.3 Solve Linear Systems by Adding or Subtracting
Example 1: Solve the linear system by adding or subtracting.
2x + 4y = 2
4x – 4y = 16
Equation 1
Equation 2
Example 2: Solve the linear system by adding or subtracting.
7x + 5y = 18 Equation 1
7x – 3y = 34 Equation 2
Example 3: Solve the linear system by adding or subtracting.
6x – 4y = 10 Equation 1
13y = 6x + 8 Equation 2
Example 4: A riverboat travels 28 miles upstream in 7 hours. It travels 28 miles downstream in 5
hours. The speed of the current remains the same during the travel upstream and downstream.
Find the average speed of the riverboat in still water and the speed of the current.
Lesson 6.4: Solve Linear Systems by Multiplying First
Least Common Multiple: _________________________________________________________
______________________________________________________________________________
Example 1: Solve the linear system.
3x – 2y = –4
7x – 4y = –6
Equation 1
Equation 2
Example 2: Solve the linear system.
5x +2y = –18
7y = 3x + 19
Equation 1
Equation 2
Lesson 6.5: Solve Special Types of Linear Systems
Inconsistent system: _____________________________________________________________
______________________________________________________________________________
Consistent dependent system: _____________________________________________________
______________________________________________________________________________
Example 1: Show that the linear system has no solution.
–5x + 4y = 16 Equation 1
5x – 4y = 8
Equation 2
Example 2: Show that the linear system has infinitely many solutions.
2
y = x +5
Equation 1
3
–2x + 3y = 15 Equation 2
Example 3: Without solving the linear system, tell whether the linear system has one solution, no
solution, or infinitely many solutions.
7x – 2y = 9 Equation 1
7x – 2y = –1 Equation 2
3x + y = –10 Equation 1
–6x – 2y = 20 Equation 2
Example 4: A pizza parlor fills two pizza orders. Find the cost of one medium pizza.
Medium
4
8
Large
12
24
Cost
$168
$336
Number of solutions of a linear system
Number of
Solutions?
What are the
lines doing?
Characteristics
of the lines
Lesson 6.6 Solve Systems of Linear Inequalities
System of linear inequalities: ______________________________________________________
______________________________________________________________________________
Solution of a system of linear inequalities: ___________________________________________
______________________________________________________________________________
Graph of a system of linear inequalities: _____________________________________________
______________________________________________________________________________
Example 1: Graph the system of inequalities.
y<
1
x+2
Inequality 1
y  –2x + 5
Inequality 2
2
Example 2: Graph the system of inequalities.
y 5
x<4
y  2x + 2
Inequality 1
Inequality 2
Inequality 3
Example 3: Write a system of inequalities for the shaded region.
Example 4: A logo contest requires that the logo width be between 3 and 5 inches, the height no
less than 2 inches and the sum of the width and height no more than 9.
a. Write and graph a system of linear inequalities that describes the information given.
b. You enter a logo 4 inches wide and 6 inches high. Use the graph to determine if it meets
contest requirement.