Download Graphical Solutions to Linear Systems Intersecting Lines Coinciding

Graphical Solutions to Linear Systems
Intersecting Lines
Parallel
One Solution
Coinciding Lines
No Unique Solution
(All reals on line)
Dependent
Independent
Slope:
y-intercept:
No Solution
Inconsistent
Slope:
y-intercept:
Slope:
y-intercept:
Classify the system without graphing:
a)
3x + y = 5
15x + 5y = 2
b)
y = 2x + 3
-4x + 2y = 6
c)
x–y=5
y + 3 = 2x
3-1 Page 161 #12-14
Without graphing, classify each system as independent,
dependent or inconsistent.
12.
6x + 3y = 12
y = -2x + 4
13.
y = -x + 5
x – y = -3
14.
x + 2y = 2
y = -0.5x - 2
Algebra 2
Section 3.1 Name____________________
SOLVING LINEAR SYSTEMS USING A
CALCULATOR
1
X − 3.
3
To do this we must find where the two lines
Ex.) Solve the system: Y = X + 2 & Y =
INTERSECT!
1)
Y=
2) Enter the first equation as Y1 = .
Enter the second equation as Y2 = .
3)
2ND
TRACE
Don’t forget parentheses!
(CALC)
4) Choose 5 which is intersect.
5) Use arrow keys to move the cursor to the approximate point
of intersection. Screen will say "First Curve?" ENTER
6) Screen will say "Second Curve?" ENTER
7) Screen will say "Guess?" ENTER
8) Screen will say “Intersection”
X = −7.5
Y = −5.5
Write the answer as an ordered pair to
Answer: ( -7.500, -5.500)
(I know, I know, …but it’s good practice for the future!)
3 decimal places!
H.W. #6
Name_____________________________
Alg. 2 (End of Ch. 3)
DO YOU REMEMBER? #1
(Review of Chapters 1 & 2)
Show all your work or no credit!
From Section:
1) Evaluate: (6 + 4y) ÷ (4x) when x =
2) Solve:
3) Solve:
1
and y = 2.
2
(1.2)
5(3 − 5x) = 6 − (3 + x)
1 − 2x ≤ 7
4) Solve for a:
ab − 6ab2 = − 3
5) State the property illustrated: ⅓ (3) = 1
(1.3)
(1.7)
(1.5)
(1.1)
6) Find the x & y-intercepts. (Write your answers as ordered pairs.)
− 5x + 10y = 4
(2.3)
DYR #6 ch 3
7) Write an equation of the line which passes through
1
(3, − 1) and which has a slope of − .
2
8) Write an equation of the line which passes
through (3, − 2) and (2, − 1).
(2.4)
(2.4)
(2.5) 10) Graph: y = − x + 3 + 1
9) Graph: x − 4y < 8
(2.6)
ANSWERS
1) 7
2)
1
2
3) − 3 ≤ x ≤ 4
6) (-4/5, 0 (0, 2/5) 7) y = −
9)
y>
1
1
x+
2
2
1
x −2
4
4) a =
−3
b − 6b 2
8) y = − x + 1
10)
5) Inverse of Mult.
H.W. #1
Name_______________________
Alg.2 SECTION 3.1 WS &
P.
SOLVING LINEAR SYSTEM WITH A GRAPHING CALCULATOR
(ROUND ALL ANSWERS TO 3 DECIMALS.)
3
1
X−
4
2
1) Y =
Y= −
Y=
4Y = 22
7) 5X + 16Y = 15
=1
2
X+1
11
3) Y = −2X + 2
Y=
4) Y =
5X +
5
8
X+
6
3
5
1
X−
12
3
2) Y =
6) 7X − 8Y = 5
1
13
X−
2
2
1
X+3
22
8) 1.7X + 3.5Y = 14.2
−3.6X − Y = 8.9
9) Y = 7.98
2.4X + 3.7Y = 14
Y = −4 X − 5
5) Y = 3X + 11
Y=
1
19
X+
3
3
10) 2.4x -1.3 - 5.6Y = 14
Y = −6X + 12
WS#1
ANSWERS
1) (2.000,1.000)
2) (5.677, 2.032)
3) (3.400, -4.800)
4) (-1.978, 2.910)
5) (-1.750, 5.750)
6) (2.882, 1.897)
7) (-6.333,2.917)
8) (-4.161,6.078)
9) (-6.469, 7.980)
10) (2.291, -1.750)
Now do Book work Page #
Alg. 2 Chapter 3 Review
ANSWER KEY
1a) (-1,-2)
b) (2, 3)
6a) (.309, 1.044)
b) (-1.364, 2.091)
2a) None, lines are parallel
b) Infinite, lines coincide
c) One, lines intersect
3a) YES
b) NO
7a) (-4,0)
b) (-1,5)
c) (3,-18)
8) 298 letters
4) (2, -1)
5) (-1, 3)
9a)
b)
c)
Vertices:(-2,
2)
( 2, 2 )
( 0, -2 )
Algebra 2
Name_________________________
CHAPTER 3 REVIEW
1) Solve the system by GRAPHING:
a) x − y = 1
Ans. (
2)
2x + y = − 4
,
b) − x + 2y = 4
)
− 2x + y = − 1
Ans. (
,
)
How many solutions to the system?
a) x + 3y = 3
x + 3y = − 3
b) 5x + 2y = − 4
− 10x − 4y = 8
c) x + 3y = 1
7x – y = 13
Ans. _____________
Ans. _________
Ans. _______
Chapter 3 Review
3) Is (-2, -1) a solution of the system?
a) 7x + 2y = –16
–6x + 9y = 3
Ans. _______
b) 3x – y = –5 –4x + 2y = 14
Ans. _______
4) Solve by SUBSTITUTION:
2x – 5y = 9
Ans: (
,
y = 3x – 7
)
5) Solve by LINEAR
2X + 3Y =
7
4X – 2Y = – 10
Ans: (
,
)
COMBINATIONS (Elimination):
Chapter 3 Review
6) Solve by using your GRAPHING CALCULATOR.
(Round your answer to 3 decimal places.)
1
a) y =
y = 5x −
b) y = –3x – 2 y =
2
Ans: (
,
)
Ans: (
,
)
7) Use SUBSTITUTION or LINEAR COMBINATIONS. Your choice!!!!
a) –2x + 3y = 8
x – 5y = –4
(
,
)
b) 5x – 2y = –15
7x + 5y = 18
(
,
)
c) 6x + y = 0
15x + 2y = 9
(
,
)
Chapter 3 Review
8) A company paid $262.26 to mail 668 letters. Some letters needed $0.29
postage and others needed $0.52 postage. How many letters were mailed at
the $0.52 rate?
Ans: ______________
9) Graph the system of linear inequalities.
a) y< 2 x>1
b) 3y ≤ x
2x – y < 0
c) y> 2x – 2
y > -2x –2
y<2
Vertices: (
,
)
(
,
)
(
,
)
3.3 Solving Systems of Linear INEQUALITIES
To graph the system:
a) Sketch the line that corresponds to the border of ONE inequality.
Don’t forget to make the line dotted for ____________ & solid for _________.
b) Draw arrows pointing in the direction you should shade.
c) The “SOLUTION” to the system is the region where all of the shaded
regions
_____________________. Darken in this region!
1) y> x + 1
y< 3
2) –y < 1
y < 2| x - 5 | - 4
3-3 (continued)
3) y< 3
y> -x + 1
x< 4
Vertices:____________________________
4) y ≤ 1
x − 2y < 4
1
x + y > −2
2
Vertices:_________________
3.1 & 3.2 Problem Solving Using Linear Systems
a) Choose a variable to represent each unknown.
b) Write an equation to represent each condition.
c) Solve the system & check your answers.
Ex. 1) You purchase 10 bags of balloons & 6 rolls of crepe paper for $20.10.
Later you decide you need more for your decorations & go back and
purchase 4 bags of balloons & 8 rolls of crepe paper for $12.80.
What was the price of each item?
Let _____ = Price of a bag of balloons
Let _____ = Price of a roll of crepe paper
Check
Ans: ________ per bag of balloons
________ per roll of crepe paper
3-1 & 3-2 (continued)
Ex. 2) You and a friend share the driving on a 280-mile trip. Your average
speed is 58 mph. Your friends average speed is 53 mph. You drive for one
hour longer than your friend. How many hours did each of you drive?
Let _______ =
Let _______ =
Answer:_________________________________________________
Ex. 3) A sporting goods store receives a shipment of 124 golf bags. The
shipment includes 2 types of bags, full-size and collapsible. The full-size
bags cost $38.50 each. The collapsible bags cost $22.50 each. The bill for
the shipment is $3,430. How many of each type of golf bag are in the
shipment?
Answer:________________________________________________
H.W. #3:
WORKSHEET & P.
Name___________________________
Algebra 2
3.1& 3.2
Ex. 1) A coin bank contains 30 coins, all dimes & quarters, worth $5.70.
How many dimes & how many quarters are in the bank?
Let _____ = number of _________________
Let _____ = number of _________________
Ans: ________ dimes
________ quarters
2) On Monday you buy 6 chicken sandwiches & 6 beef sandwiches for $30.
Then on Tuesday you go to the same deli & buy 4 chicken & 8 beef
sandwiches for $30.60. What is the price of each sandwich?
Ans:
WS#3 (continued)
3) Derado worked a total of 57 hours on two jobs. One job paid $4.50 an
hour and the other paid $4.75 an hour. His total pay for the 57 hours was
$260.75. How many hours did he work at each job?
4) Three hundred and forty-three tickets were sold to a play.
Eighty-five more student tickets than adult tickets were sold.
How many tickets of each type were sold?
Answers:
1) 12 dimes, 18 quarters
2) $2.35/chicken, 2.65/beef
3) 40 hrs. @ $4.50, 17 hrs. @ $4.75
4) 214 student tickets, 129 adult tickets
3.2 Solving Linear Systems ALGEBRAICALLY
SUBSTITUTION METHOD:
1) Solve one of the equations for x or y. Look for a variable
with a coefficient of 1 or –1.
2) Substitute this expression into the other equation & solve.
3) Substitute the value in the revised 1st equation & solve.
4) Write your answer as an ordered pair.
5) Check your answer.
Solve each by substitution:
1)
–2x + y = 8
y = –3x – 2
(Notice: This equation is already
__________________________!)
Check
–2x + y = 8
2
Answer: (
2)
,
8x + y = 12
Answer:
)
–2x + 3y = 10
y = –3x –
3-2 (contined)
3) 5x – 3y = 2
x + 2y = 3
Answer :
ELIMINATION or
LINEAR COMBINATION METHOD:
1) Multiply each term of one (or both) equation by a number to
get OPPOSITE COEFFICIENTS! (Ex. 3x & -3x, -4y & 4y)
2) Add the equations. (One variable will drop out!) Solve.
3) Substitute into the 1st equation to get the other variable.
4) Use the 2nd equation to check.
5) Write your answer as an ordered pair.
Ex. 4) 11X + 7Y = 9
6X + 7Y = 24
3-2 (continued)
Solve by elimination or linear combinations continued….
5)
6X + Y = –5
4X – 3Y = –7
Answer:
6)
3X – 5Y = –6
–2X + 7Y = 4
3-2 Continued)
7)
3X – 2Y = 5
–6X + 4Y = 7
Answer:
8)
3X – 2Y = 5
–3X + 2Y = –5
Answer:
Important Note:
If the variables “drop out” and the equation you have left is
FALSE, (Ex. 0 = 3), then the answer is ____________.
If the variables “drop out” and the equation you have left is
TRUE, (Ex. 7 = 7), then the answer is _____________.
Chapter 3 SYSTEMS OF LINEAR
EQUATIONS & INEQUALITIES
3.1 Solving Linear Systems by Graphing
A system of linear equations will have:
a) Exactly ONE solution if the graphs of the lines ___________.
Note: The slope of the lines will be ________________.
b)
NO solution if the graphs of the lines ____________________
Note: The slope of the lines will be ___________________
& the y-intercepts of the lines will be _________________.
c) An INFINITE number of solutions if the lines ____________.
Note: The slope & the y-intercepts of the lines will be _________.
Graph each “system” (both equations) on the same coordinate plane:
1) Y =
2) Y =
Y=
Y=
Lines _______________.
Lines ______________.
________
_________
solutions
solutions
3) Y =
Lines________.
____________
solutions
3-1 (continued)
To determine if an ordered pair is a solution to a system, ________________
in BOTH equations. The answer is YES, the ordered pair IS a solution, only if:
_________________________________________________________________.
Ex. 1) Is (1, 3) a solution to the system?
5X – 3Y = –4
5( ) – 3( ) = –4 ?
X + 2Y = 6
( ) + 2( ) = 6?
Answer:_______ because _______________________________________.
Ex 2) Determine if (1, 4) is a solution to the system:
−5X + Y = –1
–3X + Y = 1
y
Ex 3) Graph the system of equations:
−5X + Y = –1
–3X + Y = 1
The solution to the system is (
,
).
SUMMARY
Number of
solutions
0
1
infinite
Lines :
Slopes are:
Y-intercepts are:
Algebra II - Linear Systems
Section 3.1& 3.2 Application
Name _______________
Write a verbal model for these problems.
Assign labels to the verbal model.
Use the labels to write a linear system that represent the problem.
Solve the system and answer the question.
1. A grain storage warehouse has a total of 30 bins. Some hold
20 tons of grain each, and the rest hold 15 tons each. How many
of each type of bin are there if the capacity of the warehouse is
510 tons?
2. A caterer's total cost for catering a party includes a fixed cost,
which is the same for every party. In addition, the caterer
charges a certain amount for each guest. If it costs $300 to serve
25 guests and $420 to serve 40 guests, find the fixed cost and the
cost per guest.
3. Tickets for the homecoming dance cost $20 for a single ticket
or $35 for a couple. Ticket sales totaled $2280, and 128 people
attended. How many tickets of each type were sold?
3.3 WS
4. The perimeter of a rectangular park is 640 yards. The length of
the rectangle is 20 yards less than twice the width. What are the
length and width of the park?
5. Two isosceles triangles have the same base length. The legs
of one of the triangles are twice as long as the legs of the other.
Find the lengths of the sides of the triangles if the perimeters are
23 inches and 41 inches.
6. Friday, Music Land sold CDs at $25 each and cassette tapes at
$18 each. Receipts for the day totaled $441. On Saturday the
store priced both items at $20, sold exactly the same number of
each item, and had receipts of $420. How many CDs and cassette
tapes are sold each day?
Answers: 1) 12,18 2) $8, $100 3) c 56, s 16 5) 9,18,5 6) 12, 9 cd
Algebra II - Linear Systems
More Section 3.1 & 3.2
Name _______________
Write a verbal model for these problems.
Assign labels to the verbal model.
Use the labels to write a linear system that represent the problem.
Solve the system and answer the question.
1) A purse contains 21 coins, in nickels and dimes. The
total value is $1.65. How many coins of each kind are
there?
2) The sum of a number and twice a second number is
56. The second number is 7 less than triple the first.
Find the numbers.
3) Jake invested $10,000, part at 6% and the rest at 5%.
The total yearly investment income is $566. Find the
amount of each investment.
Answers: 1) n=9,d=12 2) 10, 23 3) $6600 at 6%, $3400 at 5%
4)28, $5;72, $8 5) 15 boat, 3 current 6) 120yards by 200 yards
More 3.1 & 3.2
4) One kind of nuts sells for $5 per pound and another
kind of nuts sells for $8 per pound. How many of each
kind should be used to make a mixture of 100 pounds
worth $716?
5) A boat travels 36 miles down stream in 2 hours. It
takes 3 hours to travel the same distance going
upstream. Find the rate of the boat in still water and the
rate of the current.
6) The perimeter of a rectangular park if 640 yards. The
length of the rectangle is 40 yards less than twice the
width. What are the length and the width of the park?
Warm Up Solving Systems
Algebra 2 Section 3.1
Name _______________
Solve using your calculator.
1)
3.5x + 4.9y = 2.1
6.8x + 2.9 y = -12
Solve by substitution.
2) 2x -3y = 13
3x + y = 3
How many solutions are there and explain why (use
slopes and y-intercepts)?
3) 3x - 4y = 2
3x - 4y = -2
4)
–x + y = 3
2x – 2y = -6
Answers: 1) (-2.8006,2.4290) 2) (2,-3) 3) zero 4) many 5) one
5) 2x + 2y = 3
2x – y = 4
Warm Up Solving Systems
Algebra 2 Section 3.2
Name _______________
Solve using appropriate method.
1)
6x + y = -5
4x – 3y = -7
3) –2x – 4y = 2
10x + 20y = -10
2)
9x + 12y = 3
3x + 4y = -2
4)
x–y=4
-3x + y = 4
Answers: (-1,1); no solution; all solution on line; (-4, -8)
Warm Up 3.1 and 3.2
Algebra 2
Name _______________
1) Is (2,3) a solution to the system?
4x + 5y = 23
6x – 3y = 0
2) How many solutions (1,0 or infinitely many) and why?
a) Y = 3x + 4
Y = 3x + 5
b)
y = ¼x - 2
y = .25x - 2
3) Solve by substitution.
5)
y = 5x + 1
y = 7x + 1
4) Solve by Linear Combination.
3x + y = 5
5x + 2y = 7
3x + 7y = 5
5x – 2y = 22
5) Solve by graphing. Use your calculator and round to 3
decimal places.
a) 5x – 3y = 7
x + 6y = -4
b)
2.4(x - 1.3) + 5.8y = 14
y = -6x +11
Answers: 1) no 2a) zero 2b) inf many 2c) one 3) (3,-4) 4) (4,-1) 5) (.909,-.818) (1441,2.356)
6) (3,1) 7) (2,-1) 8) (3,4) 9) (3,-2)
3.1 and 3.2 Warm Up Page 2
6) Graph the systems and find the solution.
6) x – y = 2
2x + 3y = 9
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
7)
4x + 2y = 6
3x - 4y = 10
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
8) 2x – y = 2
-2x + 3y = 6
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
9)
4x – 3y = 18
6x + 9y = 0
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
Name _____________
Graph, state the solution and what type of lines they are.
More Practice 3.1
1) 2x – y = -5
x + 2y = 0
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
2)
-2x + 3y = 12
2x – 3y = 6
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
3) 2x – y = 5
-4x + 2y = 10
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
......................
4)
....
....
....
....
....
....
....
....
....
....
....
....
-x + 2y = 4
-3x + 4y = 4
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
Algebra Chapter 3 Worksheet #
Name _________________
In exercises 1 to 5, use substitution to solve the linear system.
1)
2x + y = 9
3x – 4y = 8
4)
2x + y = -9
3x + 5y = 4
2)
3x + 5y = 12
x + 4y = 11
5)
3)
x – 9y = 25
6x – 5y = 3
-x + 3y = 18
4x – 2y = 8
In exercises 6 to 12, use linear combination to solve the linear
system.
6)
-2x + 7y = 10
x – 3y = -3
7)
2x – y = 2
-5x + 4y = -2
8)
3x + 11y = 4
-2x – 6y = 0
Algebra Chapter 3 Worksheet #
9)
-9x + 5y = 1
3x – 2y = 2
(continued)
10) 7x + 20y = 11
3x + 10y = 5
11) 2x – y = 3
9x – 6y = 6
In exercises 13 to 17, use any method to solve the linear system.
12)
x – y = 10
3x – 2y = 25
15) -4x – 10y = 12
x + 5y = 2
13)
–7x + 5y = 0
14x – 8y = 2
16) 5x + 16y = 15
-2x – 4y = 1
14)
4x + 3y = 1
-3x – 6y = 3
17) 4x + y = 2
6x + 3y = 0
In exercises 18 to 20, how many solutions does the linear system
have?
18) 3x + y = 20
19) 6x – y = 5
20) 3x – y = -3
2x + 2y = 4
12x – 2y = 3
-3x + y = 2
Answers: 1)(4,1) 2)(-1,3) 3)(-2, -3) 4) (-7, 5) 5) (6,8) 6) (9,4) 7) (2,2) 8)(-6,2) 9) (-4-7)
10) (1,1/5) 11) (4,5) 12) (5,-5) 13) (5/7,1) 14) (1,-1) 15)(-8,2) 16) (-19/3, 35/12)
17) (1,-2) 18) 1 sol, 19)none 20) none
Algebra 2 DYR
Ch. 3
Name:
WARM-UP
1) Solve:
7 = 7(2x + 5) – 6(x + 8)
2) Solve & graph:
x− 2 ≤3
Solution:
Graph:
3) Solve for P:
shown:
A = P + Prt
4) State the property
5) Find the x-intercept of:
line
−2x + 3y = 7
6) Write the equation of the
7) For y = − 3x + 6 − 2 state:
8) Graph x − 2y ≥ 4
a) vertex ____________
b) opens ________
c) “slope” _________
3 + -3 = 0
through (4, 11) and (5, 9).
More Section 3.3 Solving Systems of Linear Inequalities
Graph the system.
⎧ y > x + 1
1. ⎨
⎩ y ≤ 3
⎧− y < 1
2. ⎨
⎩ y ≤ 3
Vertices:____________________
Vertices: ____________________
⎧ y < 3
⎪
3. ⎨ y ≥ − x + 1
⎪ x < 4
⎩
⎧
⎪
⎪⎪ y ≤ 1
4. ⎨ x − 2 y < 4
⎪ 1
⎪ x + y > −2
⎪⎩ 2
Vertices:____________________
Vertices:_________