Name ______________________
Date _______________
Algebra 2B - Solving Systems of Equations Graphically
Block #______
WS# 2-1
Do Now: Find the slope(m) and the y-intercept(b) of each equation below.
a) y = -2x + 3
d) 5x + 6y = 12
b) 2y = 3x -6
e) y - 3 = 3(x - 4)
c) 2x – y = 4
NOTES: Using the graphs below, find the solution to each system of equations (there are two lines for
each graph!
Steps to solving a system of equations graphically:
Solve each system of equations graphically
1.
𝑦 = −2𝑥 + 7
𝑦 = 3𝑥 − 8
2.
y
𝑦 = −3𝑥 − 1
𝑦=𝑥+7
y
x
x
Solution:
Solution:
Check:
Check:
3.
𝑥+𝑦=8
𝑥 − 𝑦 = −2
4.
2𝑥 + 𝑦 = 1
1
𝑦 + 3 = − (𝑥 + 4)
2
y
y
x
x
Equations in Slope-Intercept Form:
Equations in Slope-Intercept Form:
Solution:
Solution:
Check:
Check:
Name ______________________
Date _______________
Algebra 2B - Solving Linear Systems Graphically - Homework
1.
1
𝑦 = 𝑥+3
3
𝑦= 𝑥+ 3
2.
2
Block #______
WS#2-1A
2
1
𝑦 = 2𝑥 − 6
𝑦 = 𝑥−7
4
y
y
x
3.
x
Solution:
Solution:
Check:
Check:
3
𝑦 =− 𝑥+2
4.
2
1
2
𝑦= 𝑥−3
3
4
𝑦= 𝑥+6
𝑦= 𝑥+2
2
6
y
y
x
x
Solution:
Solution:
Check:
Check:
3.
2𝑥 − 𝑦 = 8
𝑥 =𝑦+2
4.
𝑦 − 3 = 2(𝑥 + 1)
𝑦 = −3(𝑥 + 5)
y
y
x
x
Equations in Slope-Intercept Form:
Equations in Slope-Intercept Form:
Solution:
Solution:
Check:
Check:
Name ______________________
Date _______________
Algebra 2B - Solving Linear Systems Graphically (with Calc!)
DO NOW:
Solve the following system of equations graphically:
y=5
x = -2
Block #______
WS# 2-2
y
x
NOTES:
Steps to solving a system of linear equations graphically with the calculator:
1) Solve the following system of equation by graphing each on the same coordinate plane. Round your
answer to the nearest hundredth.
y=x+3
y
3y + x = 6
x
2) Solve the system graphically. Round your answer to the
nearest hundredth.
y – 2x = 3
x+y=7
y
x
3) Solve the following systems of equation graphically. Round
your answers to the nearest tenth.
a. x – 2y = -10
4x + 4y = 24
y
x
b. 3x + 4y = 12
2x – 3y = -6
y
x
4) Solve the following systems. Round your answers to the nearest tenth.
a. y= x + 4
b. 5x + y = -4
y + 3x – 2
x – y = -2
y
y
x
x
Name ______________________
Date _______________
Algebra 2B - Solving Linear Systems Applications Graphically
1) Solve the linear system. Round your answers to the nearest tenth.
y = 4x -1.5
y + 2x = 1.5
Block #______
WS# 2-3
y
x
2) A business rents in-line skates and bicycles. The in-line skates rent for $15 per day and the bikes rent for
$30 per day. During one day, the business has a total of 25 rentals and collects $450 for the rentals.
a. Write the equations (there are two!) that represent the situation above. Remember to define your
variables.
b. Find the number of pairs of skates rented and the number of bicycles rented. (You must solve the
system graphically)
c. Explain what your answer means in the context of the problem.
3) It costs $15 for a yearly member to a movie club at a movie theater. A movie ticket costs $5 for club
members and $8 for non-members .
a. Write a system of equation that you can use to find the number x of movies viewed for both
members and non-members.
b. Graph the system of equation and find where the cost for members and non-members is the same.
c. What circumstances does it make sense to become a movie club member? Explain using your graph.
4) Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small
pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold?
a. Write a system of equations to represent the situation above. Remember to define your variables.
b. Solve the system graphically.
c. Explain within the context of the problem
what the solution means.
Name _____________________
Date ___________________ Block #___________
Algebra 2B - Solving Systems by the Substitution Method
Worksheet #2-4
1.
Solve the following systems of equations by the substitution method. Show your work.
a.
b.
c.
8x + 5y = -14
y = -3x
6x - 4y = 38
x+y=5
x - 3y = 9
6x - 5y = 2
Steps for Substituion:
Now complete the systems of equations on the board on the back of the sheet.
 y  2x
1. 
4 x  y  6
2 x  3 y  4
2. 
x  2 y  1
3x  2 y  7
3. 
 x  2 y  3
3x  y  5
4. 
4 x  2 y  10
Name _____________________
Date ___________________
Block #___________
Algebra 2B - Solving Systems by the Substitution HOMEWORK
Worksheet #2-4A
Solve each system of equations using the substitution method.
1.
3 x  y  15
y3
2.
4a  b  20
a4
3.
3 x  4 y  11

2 x  y  0
4.
5 x  y  5

9 x  4 y  20
5.
2 x  3 y  2

3x  y  4
6.
3 g  2h  12
 g 9 h
7.
The perimeter of Mrs. McCord's rectangular garden is 100 feet. This can be represented with the equation 2w  2l  100 , where
w is the width of the garden (in feet) and l is the length of the garden (in feet). If the length of the garden is 1.5 times longer
than the width ( l  1.5w ), what are the dimensions of the garden?
8. Is
(5,6) a solution to the following system? How do you know?
3 x  2 y  27
x  y 1
9. Graph the following system of equations and find the solution. Remember to put your equations into
Slope- intercept form!!
x  3y  1

 x  y  1
y
x
Name _____________________
Date ___________________
Algebra 2B – Solving systems of equations using Elimination
Block #___________
Worksheet # 2-5
Do Now: Solve the following systems of equations using substitution.
x – 3y =0
3x + y = 10
There is another way to solve systems of equation algebraically – using the method of
ELIMINATION….
When to use elimination:
(Problems 1–3) In each of these systems, rewrite the equations so that it “lines up” with the first one.
1.
3x + 2y = 10
2.
4x + 2y = 6
3.
y = 2x – 9
3y – 2x = 25
x + 10y = 0
–15 = 5y + 2x
(Problems 4–6) Once the equations are lined up, you can sometimes eliminate a variable by adding the two equations together.
4.
2x – y = –2
5.
x + 2y = 9
6.
4x – 2y = 16
3x + y = 17
–2x + y = –8
– 4x + 8y = 8
(Problems 7–9) Sometimes you can eliminate a variable by multiplying one of the equations by the same number on both sides and
then adding it to the other.
7.
2x + y = 2
8.
–x + 2y = –7
9.
x + 3y = 5
3x – 2y = –11
3x + 5y = –1
2x + 4y = 7
*How would you solve this system? This one is a little more difficult.
2x + 5y = 45
3x – 2y = 20
Steps to solving a system of linear equations using ELIMINATION:
(Problems 10–20) Complete the solutions to the systems
10.
2x – y = –2
11.
x + 2y = 9
3x + y = 17
–2x + y = –8
12.
4x – 2y = 16
– 4x + 8y = 8
–x + 2y = –7
3x + 5y = –1
15.
x + 3y = 5
2x + 4y = 7
13.
2x + y = 2
3x – 2y = –11
19. 2x + 5y = 45
3x – 2y = 20
14.
20. 2x + 3y = 30
–3x – 4y=-41
Name _____________________
Date ___________________
Algebra 1 – Selecting an Algebraic Method
Do Now: Fill out the table below using the systems below.
Block #___________
Worksheet # 2-6
I would choose substitution for # ___ and #_____ I would choose elimination for # ___ and #_____
because ……
because ……
Solve each system of equations using either the substitution or elimination method. Explain your choice for each
system.
1.
2x -3y =12
x = 4y+1
2.
2x – y = 11
8y – 2x = 52
3.
8x + 14y = 4
-6x - 7y = -10
4.
8x + 8y = 24
x = –5y + 11
For each problem, write a system of linear equations and solve.
1.
The perimeter of a rectangle is 94 centimeters. The length is 10 centimeters longer than the width. Find the
dimensions (dimensions means the length and width) of the rectangle.
2.
Two numbers differ by 71. If twice the lesser number is added to the other number, the result is 500. Find the
numbers.
Extra practice:
Name _____________________
Date ___________________
Algebra 1 – System of Equations Word Problems
Block #___________
Worksheet #2-7
Break-Even Analysis: Popcorn
The business club is going to sell popcorn at hockey games. Since they are astute business men and business
women, they know that they will not make a profit right away because they have to pay the cost of buying a popcorn
machine. They need to know how many bags of popcorn they must sell in order to cover the set-up costs. In other
words, what is the break-even point for their popcorn business?
Here is what they know
 The red and glass popcorn carts often seen at carnivals and fairs costs $450.
 The popcorn, butter, salt, and serving bags cost $15 for every 100 bags of popcorn.
Which is a fixed cost? Why?
Which is variable cost? Why?
What is the cost per bag for the variable costs?
What is the equation in words for the total cost for selling popcorn at the hockey game?
1.
Write an equation for the Total Cost as a function of the number of bags of popcorn made. Use the notation C
for total cost, and let x be the number of bags of popcorn they make.
Each bag of popcorn sells for $1.00. The revenue is the amount of money they receive from selling bags of popcorn.
If they sell 20 bags of popcorn, they will receive $20, since each bag sells for $1. The revenue they take in is the
price per bag of popcorn multiplied by the number of bags of popcorn sold.
What is the equation for the revenue in words?
2.
Write an equation for the Revenue as a function of the number of bags of popcorn sold. Label the revenue
function R.
The break-even point occurs when the amount of money they receive from selling popcorn is equal to the amount
of money they spent to make the popcorn. It is when Revenue = Total
how many items they must create and sell in order to recover their expenses.
Cost. The break-even point tells
3.
Take the Total Cost and Revenue functions that you developed above, and sketch the graph of the two functions
on one coordinate plane. Label the axes appropriately.
4.
Estimate the break-even point graphically.
5.
To find the break-even point algebraically, write R= C and then solve.
6.
Explain what the solution means within the context of the problem.
7. You are planning a picnic for Memorial Day. You need to buy enough hot dogs and hamburgers so that each of
your 10 guests can have two servings. You determined that hot dogs cost $0.40 each and hamburgers cost $0.80
each. You have $12 to spend on the hamburgers and hot dogs. How many hot dogs and how many hamburgers
should you buy?
3.
Your family is planning to take the Amtrak train from Hartford to New York City for a day trip. As a result of
some research you learn that your friend Jackie took the train with a group of 3 adults and 5 children and it cost
them $269.50. A cousin also took the train to the city with a group of 2 adults and 3 children and it cost them
$171.50. Find the price of an adult’s ticket and the price of a child’s ticket.
4.
During the 2008-2009 basketball season the UConn women’s team had an incredible undefeated season (39-0)
and won the NCAA championship. Maya Moore and Renee Montgomery were the top scorers during the year
and together they scored 1,398 points. If Maya scored 110 more points than Renee, how many points did each
player score during the season?
5.
During the 2008-2009 men’s basketball season, UConn’s Hasheem Thabeet and Jeff Adrien had a total of 746
rebounds. Jeff had 30 fewer rebounds than Hasheem. How many rebounds did Hasheem and Jeff each have
during the season?
6.
At the upcoming school fair, your class is planning to raise money for a class trip to Washington, DC. You plan
to sell your own version of Connecticut Trail Mix. After doing research on the cost of various ingredients, you
find you can purchase a mixture of dried fruit for $3.25 per pound and a nut mixture for $5.50 per pound. The
class plans to combine the dried fruit and nuts to make their unique Connecticut Trail Mix that sells for $4.00
per pound. After researching the number of people who attended last year’s fair, you anticipate you will need
110 pounds of trail mix. Suppose the cost of making 110 pounds is exactly equal to the revenue from selling the
trail mix. How many pound of dried fruit and how many pounds of mix nuts were used?
Name _________________________
Date ____________________ Block # __________
Algebra 2B – Solving Systems of equation word problems algebraically
Day 1
Do Now: Solve using substitution
y - x = -3
8x + 3y = 13
Notes:
Solve each question in the space provided. Remember to define each variable.
1) The sum of two number is 33. The greater number is 3 more than the lesser number. Find both
numbers.
2) The sum of the two number is 64. The difference is 16. Find the numbers.
3) A business man buys 200 stamps for $29. Some are 13 cent stamps and some are 15 cent
stamps. How many of each did he buy?
4) Your family receives basic cable television and on movies channel for $39 a month. Your
neighbor receives basic cable and two movie channels for $45.50. What is the monthly charge
for basic cable? (Assume the each movie channel has the same monthly charge.)
5) One number is 186 larger than another. The sum of the two numbers is 800. Find both numbers.
6) Bob has a collection of cards. He has 561 in all. The number of baseball cards is twice the
number of football cards. Find the number of each type of card that he has.
7) Jim has $1.15 in nickels and dimes. He has 2 more nickels than dimes. How many of each type
does he have?
Name _________________________
Date ____________________ Block # __________
Algebra 2B – Solving Systems of equation word problems algebraically
Day 2
Do Now:
1) Solve using elimination
x + 3y = 13
x+y=5
Solve each question in the space provided. Remember to define each variable.
2) Mr. Lee buys 5 shirts and 3 ties for $34. At the same store, Mr. Daly buys 3 shirts and 6 ties for
$33. Find the price of each shirt and tie.
3) For a school play, 2 adult tickets and 5 student tickets cost $8. Four adult tickets and 3 student
tickets cost $9. Find the price of each kind of ticket.
4) At a bakery, one customer buys 5 pounds of chocolate chip cookies and 3 pounds of sugar
cookies. Another customer buys 2 pounds of chocolate chip cookies and 1 pound of sugar
cookies. What is the cost of each per pound?
5) Jack buys 3 slices of cheese pizza and 4 slices of mushroom pizza for $12.50. Grace bought 3
slices of cheese pizza and 2 slices of mushroom pizza for a total cost of $8.50. What is the cost
of one slice of mushroom pizza?
6) The cost of 3 markers and 2 pencils is $1.80. The cost of 4 markers and 6 pencils is $2.90.
What is the cost of each item?
Name ________________________________ Date ________________________
Algebra 2B – Solving and Graphing Inequalities
Do Now:
Block#____
WS#
Need Kuta of compound inequal
And on linear inequal in coor sys
Name ____________________________ Date _________________ Block #_________
Algebra 2B – New York City Cab Fares
Worksheet 2Use the following information about New York City cab fares to answer the questions below.
Initial fare............$2.50
Each 1/5 mile (4 blocks) $0.40
Each 1 minute idle.....$0.40
Additional riders….... FREE
1. Zarela, who is an actuary, travels to work at 7 a.m. Her cab travels different routes depending on
traffic, but it is usually idle for a total of 6 minutes. Write an expression to determine the cab fare,
F, Zarela will pay if she travels m miles to work.
2. Use your expression to find the cab fare if Zarela has a 1.8 mile commute to work.
3. Zarela knows her trip is 1.8 miles long. On the way home from work at 5 p.m., she never knows
how long the cab will be idle. If she has $15.00 for the cab ride home and wants to include a
$2.00 tip, how long can the cab be idle on her way home from work? Use your expression from
question (1) to answer this question.
4. Estimate how much it would cost you to take a cab ride from Grand Central Station to the Empire
State Building. Use the Internet to find the distance. Assume that the cab would be idle for 3
minutes.
Name _______________________
Date______________Block # __________
Algebra 2B – Inequalities in the Real World
Worksheet 21.
Chloe and Charlie are taking a trip to the pet store to buy some things for their new puppy. They know
that they need a bag of food that costs $7, and they also want to buy some new toys for the puppy.
They find a bargain barrel containing toys that cost $2 each.
a. Write an expression for the amount of money they will spend if they purchase a bag of food
and t toys.
b. Together, Chloe and Charlie can spend no more than $40. Use this information and the
expression you wrote in part (a) to write an inequality for finding the number of toys they can
buy.
c.
Solve the inequality and graph the solution on the number line below
d. Explain what the graph of the solution means?
2.
Valley Video charges a $15 annual membership fee plus $3 for each movie rental. Tanya puts aside
$100 for renting movies for the year. How many movies can Tanya rent from Valley Video? Use an
inequality to solve this problem. Graph your solution on the number line and explain the meaning of
your graph in a sentence.
3.
You are a salesperson at Nissan. Each month you earn $2,200 plus one-fifteenth of your sales. You
want to earn more than $4000 this month. How much must you sell this month in order to earn more
than $4000? Use an inequality to solve this problem. Graph your solution on the number line and
explain the meaning of your graph in a sentence.
4.
Joe’s car needs work. The mechanic charges $140 for parts plus $48 per hour for labor. The
mechanic said the bill will be at least $300. What is the possible number of full hours that the
mechanic will work on Joe’s car? Use an inequality to solve this problem. Graph your solution on the
number line and explain the meaning of your graph in a sentence.
5.
A popular cellular phone family plan provides 1,500 minutes. It charges $89.99/month for the first two
lines and $9.99/month for every line after that. Unlimited text messages for all phone lines costs
$30.00/month, and Internet costs $10.00/month per phone line. If a family with a $200 monthly budget
buys this plan and signs up for unlimited text messaging and Internet on each phone line, how many
cell phone lines can they afford? Use an inequality to solve this problem. Graph your solution on the
number line and explain the meaning of your graph in a sentence.
Name ________________________ Date _______________________ Block #_______
Algebra 2B – Graphing Inequality Systems
WS #2AIM: SWBAT solve a system of inequalities.
Do Now: Find the x-intercept and y-intercept of the following inequality:
2x + 3y > 12
NOTES ON GRAPHING INEQUALITIES: