Algebra I: Chapter 7 Section 1



Math I: Graphing Systems of Equations

For each of the following graphs:

1) How many times do the two lines touch?

2) Identify the actual points that they touch.

Graph #1: Graph #2: Graph #3:

Graph #4: Graph #5: Graph #6:

What is a system of equations?

What does the solution look like?

Examples: Graph each equation, find the intersection.

Example System #1:

y = x + 1

y = 2x – 5

Example System #2:

y = 4x + 4

y = 4x – 5

Example System #3:

y = 3x + 4

y = 3x + 4

[pic]

Example System #4:

y = 1/2 x – 2

y = -3/2 x + 6

[pic]

There are _____________________________ types of solutions to systems of 2 linear equations:

Key Terms about Solutions of Systems of Linear Equations:

CONSISTENT: The graphs of each equation _________________________________________.

There is at _________________________________ ordered pair that satisfies both equations.

• Independent System: the system has ___________________________________ solution.

• Dependent System: the system has ____________________________________ solutions.

INCONSISTENT: The graphs of each equation are _____________________________________.

There are ____________________________ordered pairs that satisfy both equations.

SUMMARY TABLE:

| |Intersecting Lines |Same Line |Parallel Lines |

|Graph of a System |[pic] |[pic] |[pic] |

|Number of Solutions | | | |

|Terminology | | | |

Special Case – VERTICAL LINE:

Practice Problems: Solve each system of equations by graphing.

|y = – x + 5 |y = 8 – x |

|y = x – 3 |y = 4x – 7 |

|y = – x |y = 7 + 0.5x |

|y = 2x |x = – 3 |

USE THE CALCULATOR TO SOLVE A SYSTEM OF EQUATIONS.

STEP 1: Make sure each equation is in slope-intercept form ( y = mx + b)

STEP 2: [Y =], for Y1 write your first equation, for Y2 write your second equation

STEP 3A: [GRAPH]

STEP 3B: Make sure [WINDOW] is large enough with XMAX or YMAX to see both lines.

STEP 4: Press [2ND]– [CALC (Trace)], [5:Intersect], [ENTER], [ENTER], [ENTER]

Calculator will Output: Intersection: X = #, Y = # OR an ERROR message

Math I: Graphing Systems of Equations - Part 2

INTERSECTION OF LINES DESCRIBES SOLUTIONS

• Both Equations of the system need to be in “GRAPH READY FORM” = Slope Intercept

o Don’t Round: use the actual fractions for m and b

Step #1: [pic] Y1 = Equation #1 AND Y2 = Equation #2

Step #2: [pic],[pic] = Graphs the lines in the calculator

Step #3: [pic] = Find Intersection if you can’t see lines

XMIN = Move LEFT XMAX = Move RIGHT

YMIN = Move DOWN YMAX = Move UP

Step #4: [pic],[pic], [pic]

1st Curve? [pic], 2nd Curve? [pic], GUESS? [pic]

INTERSECTION: X = # Y = # or ERROR

1) y = 3x + 3 and y = 5x - 9

GRAPH READY FORM: SOLUTION:

2) y = 4 + x and y – 2x = 1

GRAPH READY FORM: SOLUTION:

3) 2y = 15 – 3x and 3y = 6 + 2x

GRAPH READY FORM: SOLUTION:

4) y = 3x + 5 and 4y – 12x = 8

GRAPH READY FORM: SOLUTION:

CALCULATOR PRACTICE PROBLEMS:

1) y = 5x – 3

y = 7 – 5x

2) y = 3x – 2

y = 8 + 3x

3) y = 1.5x + 7

y + 9 = 4x

4) y = -2x + 5

y = 3x – 6

5) y = 8

-2y + 8x = 14

TRANSLATING WORD PROBLEMS INTO SYSTEMS OF EQUATIONS:

Example #1: Suppose the following graphs represent ticket sales to the Carolina Panthers and Charlotte Bobcats games since 2000. And we were able to find the intersection of the lines.

x = years since 2000 and

y = number of tickets in thousands

What does the solution to this system of equations represent?

Make sure to relate the SOLUTION of your system to the ORIGINAL QUESTION.

PRACTICE #1 - #3: You DO NOT have to solve right now.

(1) Create the system of linear equations that describes the following statements

(2) Determine what the intersection of the system would represent.

#1: The sum of two numbers is equal to 35. The difference of the two numbers is 21.

#2: The number of boys in the school is twice the number of girls. There are a total of 420 students in the school.

#3: The bake sale is selling cupcakes and brownies today. They sold a total of 400 baked items today. Cupcakes were sold for $1.50 each and brownies were sold for $2.00 each. The total sales was $725.

PRACTICE #4 - # 7:

(1) Set Up a System of Equations

(2) Solve for the Intersection of the System

(3) Explain the Intersection in terms of the word problem.

#4: The reading of daily newspapers is modeled by the linear equation y = –2.8x +170 and the reading of online newspapers is modeled by the linear equation y = 14.4x + 2, where x represents the number of years since 1993 and y represents the average number of hours person per year.

#5: The sales of cassette singles decreased, and the sales of CD singles increased during the 1990s. Cassette sales is modeled y = 69 – 6.9x and the CD sales is modeled by y = 5.7 + 6.3x. x represents the number of years since 1991 and y represents the sales in millions of dollars.

#6: The width of a rectangle is 2 meters less than twice its length. What are the dimensions of a rectangle that has a perimeter of 40 meters?

#7: Michael and Michelle both want to buy a car. Michelle has already saved $3500 and plans to save $445 per month until she can buy a car. Michael has already saved $2900 and plans to save $495 per month until he can buy a car. How long will it take Michael and Michelle to have saved the same amount and what is that amount?

SETTING UP SYSTEM OF EQUATION WORD PROBLEMS

1. Read the problem Carefully and Identify Any Important Information

2. Define Variables (x and y)

3. Write a system (2 equations) with variables based on the given problem.

4. What would the intersection (solution point) represent in the problem?

1) The sum of two numbers is 200 and their difference is 28.What are the two numbers?

2)100 cans and 300 bottles were bought for $450. 200 cans and 200 bottles were bought for $400. What is the cost of an individual can and bottle?

3)Arsene saves $55 per week and already has saved $270. Jose has $990 and spends $35 per week. How many DAYS until Arsene and Jose have the same amount of money?

4) There are a total of 800 students in a high school. Twice the number of upperclassmen is 98 less than the number of underclassmen. How many upper and underclassmen are in the high school?

5) A total of 10,000 king and queen sized sheets were sold by a warehouse. A king sized sheet costs $25 and a queen sized sheet costs $20. All 10,000 sheets were sold for a total of $227,000. How many queen sheets were sold?

6) The width of a rectangle is 5 times larger than the length of the rectangle. The perimeter of the rectangle is 360 units. What are the measurements of the length and width?

7) The length of a rectangle is 10 more than twice the width. The perimeter is 70 units. What is the area of the rectangle?

8) Stephan has been gaining weight at 1.5 pounds a week from his original weight of 120 pounds. Seth is losing weight at 2.5 pounds a week from his original weight of 184 pounds. How much will Seth and Stephan weigh when they are the same weight?

9) Georgia bought 3 pens and 4 notebooks for $26.50 from the bookstore. She later bought 7 pens and 5 notebooks for $41.25 from the same bookstore. How much would you have to pay for 2 pens and 2 notebooks?

10) You are able to buy 4 tacos and 6 burritos for $62.58 or you could buy 5 burritos and 7 tacos for $66.78. How much would 3 burritos cost?

-----------------------

y

x

y

x

y

x

Exp #2 – Solve by hand:

x = - 2 and y = 4x + 12

Exp #1 - Graph:

x = 4

y = -3/2x + 2

Solution:

Exp #3 – Solve by hand:

2y + 3x = 14 and x = 6

Y =

ZOOM

6: ZSTANDARD

WINDOW

2ND

TRACE: CALC

5: INTERSECT

ENTER

ENTER

ENTER

y

x

Panthers Ticket Sales

(7, 85)

Bobcats Ticket Sales

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