Monday, 2/22/10 SWBAT… solve systems of equations using the graphing method

Monday, 2/22/10

SWBAT… solve systems of equations using the graphing methodAgenda1. WU (15 min)2. Cornell notes - Systems of equations: Graphing Method (2 slides) (10 min)3. 4 examples (20 min)

Warm-Up:1. Do “Get Ready for Chapter 6” problems on the back of this week’s agenda.

DO #14 & #15 FIRST(Warning – answer to #9 is wrong)

HW#1: Graphing Method

Topic: Systems of Equations: Graphing Method

What are systems of equations? A collection of equations involving the same

set of variables. We will be dealing with two equations and two

variables.

Step 1) Write the equations of the lines in slope intercept form (solve for y.)

Step 2) Graph each line on the same graph.

Step 3) Determine the point of intersection and write this point as an ordered pair.

• If the two equations represent the same line, the system of equations has infinitely many solutions.

• If the two equations have no points in common, the system of equations has no solution.

Graph each system and determine the number of solutions that it has. If it has one solution, name it as an ordered pair. Write no solution or infinite solutions where appropriate.

x – y = 2

3y + 2x = 9Step 1: Write each equation in slope-intercept form.

x – y = 2

-x -x

-y = -x + 2

y = x – 2

3y + 2x = 9

- 2x -2x3y = -2x + 93 3 3

y x 23

3-1 -1 -1

x

yStep 2: Graph each line on the same graph

Step 3: Determine the point of intersection.

The point of intersection of the two lines is the point (3,1).

This system of equations has one solution, the point (3,1).

y = x – 2

y x 23

3

Graph each system and determine the number of solutions that it has. If it has one solution, name it as an ordered pair. Write no solution or infinite solutions where appropriate.

y xy x

32 6

1.

2. x + 3y = 3

3x + 9y = 9

y = 3/5x – 4

y = 3/5x

3.

x

yThe two equations in slope-intercept form are:

y xy x

32 6

This system of equations has one solution, the point (3,0).

The point of intersection of the two lines is the point (3, 0).

x

yThe two equations in slope-intercept form are:

y x

y x o r y x

13

1

39

99

13

1

These two equations represent the same line.

Therefore, this system of equations has infinitely many solutions.

The two equations in slope-intercept form are:

x

y

y x

y x

35

4

35

This system of equations represents two parallel lines.

This system of equations has no solution because these two lines have no points in common.

Thurs, 2/25/10

SWBAT… solve systems of equations using the substitution methodAgenda1. WU (10 min)2. Examples – Systems of equations: substitution method

Warm Up:1. Any questions from hw#1?2. Any questions from problems on back of agenda?3. Review graphing examples handout – any questions?

HW#2: Substitution method

Conclusion: Fill out the chart below:

Compare m and b

Number of Solutions

One

None

Infinite

Conclusions

Compare m and b

Number of Solutions

Different m values(b can be same or different)

One

None

Infinite

Conclusions

Compare m and b

Number of Solutions


One

Same m value, but different b values

None

Infinite

Conclusions

Compare m and b

Number of Solutions


One

Same m value, but different b values

None

Same m value and same b value

Infinite

ConclusionsDescribe the advantages and disadvantages to solving system of equations using the graphing method.

Sample Answer: Graphing clearly shows whether a system of equations has one solution, no solution, or infinitely many solutions (visual). However, finding the exact values of x and y from a graph can be difficult.

Tues, 3/2/10

SWBAT… solve systems of equations using the substitution methodAgenda1. WU (15 min)2. Examples – Systems of equations: substitution method (35 min)

Warm Up:1. Write hw in planner for the whole week!2. Use substitution to solve the system of equations.Check your answers!

y = 6 – 3x 4x + 2y = 8

HW#2:Substitution method OR HW#3: Additional practice – substitution method

Wed, 3/3/10

SWBAT… solve systems of equations using the substitution methodAgenda1. WU (10 min)2. Work on hw#2 or hw#3 (20 min)3. Quiz (15 min)

Warm Up: Use substitution to solve the system of equations. Check your answers!

y = 2x – 4 -6x + 3y = -12

HW#2:Substitution method AND HW#3: Additional practice – substitution method

Quiz – Complete on graph paper

1.) Solve the system using the graphing method:y = 2x – 4

-6x + 3y = -12

2.) Solve the system using the substitution method:(Write the answer as an ordered pair)

y = 4x + 52x + y = 17

Wed, 3/3/10

Use the substitution to solve the system of equations

y = 2x – 4 x = y – 1 -6x + 3y = -12 -x + y = -1

Thurs, 3/4/10SWBAT… solve systems of equations using the elimination using addition and subtraction method

Agenda1. WU (10 min)2. 4 Examples – Systems of equations: elimination method (20 min)3. Work on hw#3 (20 min)

1. What is the solution of the following system of equations? y = 6x – 1 and y = 6x + 4

2. Write an equation that will create a system of equations with y = 4x – 3 that has no solution?

HW#4: Elimination using addition and subtraction method

Elimination using AdditionUse the elimination method to solve the system of equations:

The sum of two numbers is 24.Five times the first number minus the second number is 12. Find the numbers.

Elimination using Subtraction

Use the elimination method to solve the system of equations:

2t + 5r = 6 2t + 9r = 22

Mon, 3/8/10

SWBAT… solve systems of equations using the elimination methodAgenda1. WU (10 min)2. Examples – elimination using multiplication3. Work on hw#5

Warm-Up: 1. Write hw in planner for the week!2. Use elimination to solve the system:

6x – 2y = 1 10x – 2y = 5

HW#5: Elimination using multiplication

Elimination using Multiplication

Use the elimination method to solve the system of equations:

5x + 6y = -8 2x + 3y = -5

Tues, 3/9/10

SWBAT… apply systems of equations Agenda1. WU (10 min)2. Review hw#53. Concept Summary – Solving Systems of Equations (15 min)4. Real-life systems examples

Warm-Up:1. The table below shows the number of car’s at Santo’s Auto Repair Shop. Santos has allotted 1100 minutes for body work and 570 minutes for engine work. Write a system to determine the time for each

service.

HW#6-Real life examples and Practice Test

Item Repairs (minutes) Maintenance (minutes) body 3 4

engine 2 2

Auto Shop QuestionsQ: What does the first equation in the system of equations

represent?A: The amount of time to perform body work repair for 3 cars

and body work maintenance for 4 carsQ: What does the second equation in the system represent?A: The amount of time to perform engine repair for 2 cars and

engine maintenance for 2 cars.Q: What will the solution to the system represent?A: The number of minutes allotted per repair and the number

of minutes allotted per maintenance.Q: If you multiply the second equation by -2, what variable

could you eliminate by adding the equations?A: The variable m.

Fill in the chart below:

Method The Best Time to UseGraphingSubstitutionElimination using Addition




Method The Best Time to UseGraphing To estimate solutions, since graphing usually

does not give an exact solution. To visually see a solution.

Substitution

Elimination using AdditionElimination using Subtraction




does not give an exact solution.

Substitution If one of the variables in either equation has a coefficient of 1.

Elimination using AdditionElimination using Subtraction





Substitution If one of the variables in either equation has a coefficient of 1 or -1.

Elimination using Addition

If one of the variables has the same coefficient, but opposite signs.






Substitution If one of the variables in either equation has a coefficient of 1 or -1.




If one of the variables has the same coefficient and the same sign.


Fill in the chart below:Method The Best Time to Use

Graphing To estimate solutions, since graphing usually does not give an exact solution. To visualize the solution.

Substitution If one of the variables in either equation has a coefficient of 1.




If one of the variables has the same coefficient and the same sign.


If none of the coefficients are 1 and neither of the variables can be eliminated by simply adding or subtracting the equations (All other times)

How a customer uses systems of equations to see what he paid

Two groups of students order burritos and tacos at Atontonilco. One order of 3 burritos and 4 tacos costs $11.33. The other order of 9 burritos and 5 tacos costs $23.56. How much did each taco and burrito cost?

How a fair manager uses systems of equations to plan his inventory

The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2,200 people enter the fair and $5,050 is collected. How many children and how many adults attended?

How a manager uses systems of equations to plan employee time

The local bakery is making chocolate chip cookies and bread. Each batch of cookies take 20 minutes to prepare and 10 minutes to bake. Each loaf of bread takes 10 minutes to prepare and 30 minutes to bake. The bakery’s management has allotted 800 minutes of employee time (preparation) and 900 minutes of oven time (baking). How many of each should be baked?

How a customer uses systems of equations to see what he paid

A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totaled $487. The second order was for 6 bushes and 2 trees, and totaled $232. The bills do not list the per-item price. What were the costs of one bush and of one tree?

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Monday, 2/22/10 SWBAT… solve systems of equations using the graphing method