3.1 NOTES Solving Systems of Linear Equations Graphically A system of two linear equations in two variables x and y consist of two equations of the following form: Ax + By = C Equation 1 Dx + Ey = F Equation 2 where the solution (x,y) satisfies both equations. Checking Solutions of a Linear System: 3x – 2y = 2 x + 2y = 6 1.) Is (2,2) a solution of the above system of equations ? 2.) Is (0,-­‐1) a solution of the above system of equations ? Solving a System Graphically : Graphically, the solution of the system of equations is the point or points where the two lines intersect. Find the solution of the following system of equations graphically: 3x – 2y = 2 x + 2y = 6 Verify this answer on your graphing calculator using the intersect function. Create a table of values for each equation. 3x – 2y = 2 x + 2y = 6 x y x y 10
8
6
4
2
– 10
–5
5
–2
–4
–6
–8
– 10
10
1.) Solve the following system of equations graphically. Verify your answer on your graphing calculator. 2x – 2y = -­‐8 2x + 2y = 4 Check Algebraically. . 2.) Solve the following system of equations graphically. Verify your answer on your graphing calculator. 3x – 2y = 6 3x – 2y = 2 3.) Solve the following system of equations graphically. Verify your answer on your graphing calculator. 2x – 2y = -­‐8 -­‐2x + 2y = 8 10
8
6
4
2
– 10
–5
5
10
5
10
5
10
–2
–4
–6
–8
– 10
10
8
6
4
2
– 10
–5
–2
–4
–6
–8
– 10
10
8
6
4
2
– 10
–5
–2
–4
–6
–8
– 10
How many solutions for a system of linear equations?? Application : You are checking out cell phone plans and discover that Talk Anytime Wireless charges $50.00 per month for the first phone line and charges $20.00 per additional phone line. Text Away Wireless charges $80.00 per month for the first phone line and $5.00 per additional phone line. Use your graphing calculator to create the graph of a system of equations to determine the number of additional phone lines for which it would be cheaper to use Talk Anytime verses Text Away. 3.1 HOMEWORK The graph of a system of two linear equations is shown. Circle the phrase that applies. No Solution No Solution No Solution Infinitely many solutions Infinitely many solutions Infinitely many solutions Exactly 1 Solution Exactly 1 Solution Exactly 1 Solution Without the use of a graphing calculator, graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically. 1) x = 5 x + y = 1 2) y = -­‐5 – x x + 3y = -­‐15 10
10
8
8
6
6
4
4
2
2
– 10
–5
–2
–4
–6
–8
– 10
5
10
– 10
–5
5
–2
–4
–6
–8
– 10
10
3) 3
x + y = 5 4
3x + 4y = 2 – 10
5) – 10
4) -­‐4y = 24x + 4 y= -­‐6x – 1 10
10
8
8
6
6
4
4
2
2
–5
5
10
– 10
–5
–2
–2
–4
–4
–6
–6
–8
–8
– 10
– 10
2x – y = 7 y = 2x + 8 1
x + 7y = 2 3
2
x + 4y = 2 3
6) 10
6
4
4
2
2
5
–4
–6
–8
– 10
8
6
–2
10
10
8
–5
5
10
– 10
–5
5
–2
–4
–6
–8
– 10
10
7) You are choosing between two long distance phone services. Company A charges $.09 per minute plus a $4 monthly fee. Company B charges $.11 per minute with no monthly fee. a.) Let x be the number of minutes you call long distance in one month, and let y be the total cost of long distance phone service. Write and graph two equations representing the cost of each company’s service. (Use your graphing calculator to help you to graph the functions). b.) Use your graphing calculator to find the point where the graphs intersect. Which customers should choose Company A. Which customers should choose company B? 3.2 NOTES Solving Linear Systems Algebraically Solve the following system of equations graphically (verify your answer on your graphing calculator) 10
8
6
Substitution – 10
4
2 x − 4 y = 13
4x − 5 y = 8
Linear Combination 2
–5
5
10
–2
–4
–6
–8
– 10
Solve the system of equations graphically and then use the substitution method to solve. 10
y = 3 x − 13
2 x + 2 y = −10
8
6
4
2
– 10
–5
5
–2
–4
–6
–8
– 10
10
Solve the following system of equations by using the linear combination method. 7 x − 12 y = −22
− 5 x + 8 y = 14
Solve the following system of equations graphically and then use one algebraic method of your choice to verify your answer. 10
8
3 x − 5 y = 30
− 6 x + 10 y = −60
6
4
2
– 10
–5
5
10
–2
–4
–6
–8
– 10
Solve the following system of equations graphically and then use one algebraic method of your choice to verify your answer. 10
8
– 10
6
4
2
–5
5
–2
–4
–6
–8
– 10
10
6x − 3y = 9
4 x − 2 y = −8
Application: Cross Training You want to burn 380 calories during 40 minutes of exercise. You burn about 8 calories per minute skateboarding and 12 calories per minute running. How long should you spend doing each activity? (Hint: use two separate equations for time and calories) 3.2 Homework Solve the following problems using the substitution method. 1) 5x + 3y = 4 2) -­‐2x + y = 6 3) – y = -­‐3x + 4 y = -­‐5x + 16 4x – 2y =5 -­‐9x+3y = -­‐12 Solve the following problems using the linear combination method. 4) y = 6x + 2 5) -­‐9x + 6y = 0 6) -­‐15x – 2y = -­‐31 -­‐18x +3y = 4 -­‐12x + 8y = 0 4x + 6y – 11 = 0 Chapter 3 Word Problems – Choose to solve
algebraically or using matrices
1.) You plan to work 200 hours this summer mowing lawns and
babysitting. You need to make a total of $1300. Babysitting pays
$6 per hour and mowing lawns pays $8 per hour. How many hours
should you work at each job?
2.) You enroll in a book club in which you can earn bonus points
to use toward the purchase of books. Each paperback book you
order costs $6.95 and earns you 2 bonus points. Each hardcover
book costs $19.95 and earns you 4 bonus points. The first order
you place comes to a total of $60.75 and earns you 14 onus
points. How many of each type of book did you order?
3.) A bus station 15 miles from the airport runs a shuttle
service to and from the airport. The 9:00 AM bus leaves for the
airport traveling 30 mph. The 9:05 AM bus leaves for the airport
traveling 40 mph. Write a system of linear equations to
represent distance as a function of time for each bus. How far
from the airport will the 9:05 AM bus catch up to the 9:00 AM
bus?
4.) The school yearbook staff is purchasing a digital camera.
Recently the staff received two ads in the mail. The ad for
store #1 states that all digital cameras are 15% off. The ad for
store #2 gives a $300 coupon to use when purchasing any digital
camera. Assume that the lowest priced digital camera is $700.
When could you get the same deal at either store?
5.) One evening, 76 people gathered to play doubles and singles
ping pong. There were 26 games in progress at one time. A
doubles game requires 4 players and a singles game requires 2
players. How many games of each kind were in progress at one
time if all 76 people were playing?
6.) A citrus fruit company plans to make 13.25 lb gift boxes of
oranges and grapefruits. Each box is to have a retail value of
$21. Each orange weighs 0.5 lb and has a retail value of $0.75.,
while each grapefruit weighs 0.75 lb and has a retail value of
$1.25. How many oranges and grapefruits should be included in
the box?
7.) You are starting a business selling boxes of hand-painted
greeting cards. To get started, you spend $36 on paint and
paintbrushes that you need. You buy boxes of plain cards for
$3.50 per box, paint the cards, and then sell them for $5 per
box. How many boxes must you sell for your earnings to equal
your expenses? What will your earnings and expenses equal when
you break even?
8.) You ride an express bus from the center of town to your
street. You have two payment options. Option A: buy a monthly
pass and pay $1 per ride. Option B: pay $2.50 per ride. A
monthly pass costs $30. After how many rides will the total
costs of the two options be the same?
9.) A soccer league offers two options for membership plans.
Option A: an initial fee of $40 and costs $5 for each game
played. Option B: $10 for each game played. After how many games
will the total cost of the two options be the same?
10.) You worked 14 hours last week and earned a total of $96
before taxes. Your job as a lifeguard pays $8 per hour and your
job as a cashier pays $6 per hour. How many hours did you work
at each job?
11.) During one calendar year, a state trooper issued a total of
375 citations for warnings and speeding tickets. Of these, there
were 37 more warnings than speeding tickets. How many warnings
and how many speeding tickets were issued?
12.) A movie theatre charges $9 for an adult’s ticket and $6 for
a child’s ticket. One Friday night, the theatre sold a total of
848 tickets for $6711. How many tickets of each type were sold?
13.) In one week, a music store sold 9 guitars for a total of
$3611. Electric guitars sold for $479 each and acoustic guitars
sold for $339 each. How many of each type of guitar were sold?
14.) An adult pass for a county fair costs $2 more than a
children’s pass. When 378 adult and 214 children’s passes were
sold, the total revenue was $2384. Find the cost of an adult
pass.
15.) A nut wholesaler sells a mix of peanuts and cashews. The
wholesaler charges $2.80 per pound for peanuts and $5.30 per
pound for cashews. The mix is to sell for $3.30 per pound. How
many pounds of peanuts and how many pounds of cashews should be
used to make 100 pounds of the mix?
16.) You are on the prom decorating committee and are in charge
of buying balloons. You want to you both latex and Mylar
balloons. The latex balloons cost $.10 each and the Mylar
balloons cost $.50 each. You need 125 balloons and you have
$32.50 to spend. How many of each can you buy?
17.) A caterer is planning a party for 64 people. The customer
has $150 to spend. A $39 pan of pasta feeds 14 people and a $12
sandwich tray feeds 6 people. How many pans of pasta and how
many sandwich trays should the caterer make?
18.) For spring break you a planning a 7 day trip to Florida.
You estimate that it will cost $275 per day in Tampa and $400
per day in Orlando. Your total budget for the 7 days is $2300.
How many days should you spend in each location according to
your budget?
3.3 NOTES
Graphing Systems of Linear Inequalities
10
Graph the following system of equations: 8
2x + 3y = 6 6
x – 2y = 2 4
2
– 10
–5
5
10
–2
–4
–6
–8
– 10
Decide whether the ordered pair is a solution to the inequality listed in the table. If so, write yes. If not, write no. (0,0) (2,3) (3,-­‐1) (4,0) (-­‐3,4) (-­‐5,0) (6,0) (3,-­‐2) (1,5) 2x + 3y ≤ 6 x − 2y ≤ 2 Plot the points on your graph using the following symbols: Solution to both inequalities: * Solution to neither inequality: x Solution to exactly one inequality: o Graph the following systems of inequalities: 10
y ≥ −3x − 1
8
y<x+2
6
4
2
– 10
–5
5
10
–2
–4
–6
–8
– 10
10
8
6
x≤0
2.) y ≥ 0
x − y ≥ −2
10
8
4
2
– 10
–6
–8
– 10
−x < y
3.) x + 3y < 9 x≥2
2
–5
10
–4
4
5
–2
6
– 10
–5
5
10
–2
–4
–6
–8
10
– 10
8
6
4
2
x + 2y ≤ 10
4.) 2x + y ≤ 8 2x − 5y < 20
– 10
–5
5
–2
–4
–6
–8
– 10
10
Write the system of inequalities that correspond with the shaded region. 3.3 HOMEWORK Graph the system of linear inequalities. y > −2
y ≤1
1) 10
8
6
4
2
– 5
– 10
5
10
–2
–4
–6
–8
– 10
3) – 10
x−y>7
2x + y < 8
y > −5x
x ≤ 5y
2) 10
8
6
4
2
– 10
–5
5
–4
–6
–8
– 10
y<4
x > −3 y>x
4) 10
8
8
6
6
4
4
2
2
5
10
10
–2
10
–5
– 10
–5
5
–2
–2
–4
–4
–6
–6
–8
–8
– 10
– 10
10
2x − 3y > −6
5) 5x − 3y < 3 x + 3y > −3
– 10
6) y<5
y > −6
2x + y ≥ −1
y≤ x+3
10
8
6
4
2
–5
5
10
– 10
–5
–2
–4
–6
–8
– 10
Challenge. Write a system of linear inequalities for the region. 10
8
6
4
2
5
10
–2
–4
–6
–8
– 10
3.4 NOTES Linear Programming optimization: finding the maximum or minimum value of some quantity
linear programming: the process of maximizing or minimizing a linear objective function subject to
constraints that are linear inequalities
constraints: graphing restrictions
feasible region: the graph of the system of constraints
Ex:
Find the
minimum
value and
the
maximum
value of
the
objective
function C
= 5x – 2y
subject to
the
following
constraint
s.
x ≥0
y ≥0
2x + y ≤ 8
x + 3y ≤ 9
minimum value: ____________
maximum value: ____________
Ex: FU N DRAISER Your class plans to raise money by selling T-shirts and baseball caps. The plan
is to buy the T-shirts for $8 and sell them for $12 and to buy the caps for $4 and sell them for $7.
The planning committee estimates that you will not sell more than 120 items. Your class can
afford to spend as much as $800 to buy the articles. The constraints on your fun-raising activity
are given by the system of inequalities below. Your class can only sell combinations of T-shirts
and caps indicated by points that are solutions to the system.
c + t ≤ 120
4c + 8t ≤ 800
c ≥0
t ≥0
a) Your club wants to maximize profit. Write the profit functions p in terms of c and t.
b) At which point do you think the maximum value of p will occur?
c) Which combination of baseball caps and T-shirts maximizes profit? What is the maximum
profit?
Ex: PIN ATAS Piñatas are made to sell at a craft fair. It takes 2 hours to make a mini piñata and
3 hours to make a regular-sized piñata. The owner of the craft booth will make a profit of $12 for
each mini piñata sold and $24 for each regular-sized piñata sold. If the craft booth owner has no
more than 30 hours available to make piñatas and wants to have at least 12 piñatas to sell, how
many of each size piñata should be made to maximize profit?
3.5, 3.6 NOTES Graphing and Solving Systems of Equations in Three Variables (See Algebra In Motion Demo for Graphing in 3D)
Plot the following ordered triples:
(-1,3,4)
(3,-4,-2)
Sketch the graph of 3x + 2y + 4z = 12
Solve the following system of equations:
3x + 2y + 4z = 11
2x - y + 3z = 4
5x – 3y + 5z = -1
1.) Eliminate one of the variables in two of the original equations.
2.) Solve the new system of equations (from 1) in two variables.
3.) Substitute the known values (from 2) in to one of the original equations to solve
for the third unknown variable.
CHECK:
(
,
,
) is the point where _______________________
Solve the following system of equations:
x + 3y - z = - 11
2x + y + z = 1
5x – 2y + 3z = 21
1.) Eliminate one of the variables in two of the original equations.
2.) Solve the new system of equations (from 1) in two variables.
3.) Substitute the known values (from 2) in to one of the original equations to solve
for the third unknown variable.
CHECK:
(
,
,
) is the point where _______________________
3.5, 3.6 HOMEWORK HOMEWORK:
Plot the ordered triples in the three dimensional coordinate system.
1.)
( 2, 4, 0 )
2.)
(3, 4, -2)
3)
( -2, 1, 1 )
4)
(-3, -2, -4)
5) Sketch the graph of x + 6y + 4z = 12
6)
Sketch the graph of -18x -9y + 6z = 18
Solve the following systems of equations in 3 variables.
1.)
x + 2y + 5z = -1
2x - y + z = 2
3x + 4y - 4z = 14
2.)
5x – 4y + 4 z = 18
-x + 3y - 2z = 0
4x – 2y + 7z = 3
3.) -5x + 3y + z = -15
10x + 2y + 8z = 18
15x + 5y + 7z =9
Solve the following systems of equations in 3 variables algebraically. 1.) x + y + 2z = 1
2x − y + z = 2
4x + y + 5z = 4
2.) x + y + z = 2
x+ y−z = 2
2x + 2y + z = 4
3.) x − 2y + z = −6
2x − 3y = −7
− x + 3y − 3z = 11