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 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:_________