SWBAT… Solve a systems of equations using the graphing method Tues, 1/10

Agenda1. WU (5 min)2. Graphing method posters (20 min)3. Review HW #1

Warm-Up:

Any questions from the test corrections?

HW#1: Systems - Graphing

We are starting a new unit: System of Linear Equations

SWBAT…1. Solve a system of linear equations using the graphing

method

2. Solve a system of linear equations using the substitution method

3. Solve a system of linear equations using the elimination method (adding, subtracting, or multiplying)

4. Write and solve a system of equations based on real life scenarios

Activity-Systems of equations: Graphing

You and your partner will be given a system of equations to graph on poster board

Directions:1. Solve the system using the graphing method

2. Determine the number of solutions it has

3. If the system has one solution, name it (see worksheet on how to name it.)

4. If the system has one solution, check your answer

Step 1) Write the equations of the lines in slope-intercept form.

Step 2) Graph each line on the same graph.

Step 3) Determine the point of intersection and write this point as an ordered pair (1 solution.)

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

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

• WARNING: Extend your lines!

Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.

x – y = 2

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

x – y = 2

+ y +y

x = 2 + y

- 2 -2

x – 2 = y

3y + 2x = 9

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

y x 2

33

x

y Step 2: Graph each line on the same graph

Step 3: Determine the point of intersection.

(3,1).

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

y = x – 2

y x 2

33

Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.

1 3 3

3 9 9

.

x y

x y

23

54

5 3

. y x

y x

3 3

2 6

. x y

x y

x

yThe two equations in slope-intercept form are:

y x

y x o r y x

1

31

3

9

9

9

1

31

Plot points for each line.Draw in the lines.

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

3

54

3

5

Plot points for each line.

Draw in the lines.

This system of equations represents two parallel lines.

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

x

yThe two equations in slope-intercept form are:

y x

y x

3

2 6

Plot points for each line.

Draw in the lines.

This system of equations represents two intersecting lines.

The solution to this system of equations is a single point (3, 0).

Conclusions

The solution to a systems of equations is the point where the two lines intersect (one solution)

No solution will be parallel lines Infinite solution will be the same line

SWBAT… Solve a systems of equations using the graphing method Wed, 1/11

Agenda

1. WU (10 min)

2. Review HW#1 and graphing examples (15 min)

3. Substitution method (20 min)

Warm-Up:

1. Do the back of this week’s agenda.

HW#2: Systems-Substitution

x

y Step 2: Graph each line on the same graph

Step 3: Determine the point of intersection.

(3,1).

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

y = x – 2

y x 2

33

x

yThe two equations in slope-intercept form are:

y x

y x o r y x

1

31

3

9

9

9

1

31

Plot points for each line.Draw in the lines.

These two equations represent the same line.

Therefore, this system of equations has infinitely many solutions .

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

Yes, (1, 2) is a solution to the system.

-4 = -4 and 4 = 4

HW#1: Systems-Graphing Method Answers:

1. 1 Solution: (1, 2)

2. 1 Solution: (-4, -2)

3. Infinite Solutions

4. 1 Solution: (-2, -2)

5. 1 Solution: (-3, 5)

6. Infinite Solutions

Conclusions

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

Different m values(b can be same or different)

One

Same m value, but different b values

None

Infinite

Conclusions

Compare m and b

Number of Solutions

Different m values(b can be same or different)

One

Same m value, but different b values

None

Same m value and same b value

Infinite

Systems of equations: substitution method

Solve the system using substitutionThe sum of two numbers is 20. The difference between three times the larger number and twice the smaller is 40.Verify that your solution is correct.

Follow the directions from the box and solve the above system.

The steps using the substitution method are shown in the box on HW#2

Use substitution to solve the system of equations

1.) y = 2x – 4 2.) x = y – 1

-6x + 3y = -12 -x + y = -1

Thurs, 1/12

SWBAT… solve systems of equations using the substitution methodAgenda

1. WU (10 min)

2. Review HW#2 (15 min)

3. Quiz: Systems – graphing & substitution method (20 min)

WU: Use substitution to solve the system:

Twice one number added to another is 18. Four times the first number minus the other number is 12. Find the numbers.

HW#3: Substitution method

HW#2: Substitution Answers

1. (5, 10)2. (0, 2)3. (2, 0)4. No Solution5. Infinite Solution6. (0, -6)7. a.) t = cost of a taco, b = cost of a burrito

b.) 3t + 2b = 7.40 4t + 1b = 6.45c.) c = 1.1, b = 2.05d.) The cost of 2 tacos is $2.20 and the cost of 2 burritos is $4.10.

Fri, 1/13

SWBAT… solve a system of equations using elimination by adding or subtractingAgenda

1. WU (10 min)

2. Three examples: elimination using addition or subtraction (20 min)

3. Practice – HW#4

Warm-Up:

Negative three times one number plus five times another number is -11.

Three times the first number plus 7 times the other number is -1.

Find the numbers.

Ex.1: Elimination using AdditionNegative three times one number plus five times another number is -11.

Three times the first number plus 7 times the other number is -1.

Find the numbers.

-3x + 5y = -11

3x + 7y = -1Q: What did you notice about the x coefficients?

A: They were the opposite!

Ex. 2: Elimination using Subtraction

2t + 5r = 6

2t + 9r = 22

Q: What do you notice about the t coefficients?

A: They are the same!

Ex. 2a: Elimination using Subtraction

-2t + 5r = 6

-2t + 9r = 22

Q: What do you notice about the t coefficients?

A: They are the same!

Tues, 1/17

SWBAT… know when the best time to use each system method Agenda

1. WU: real life example using elimination w/ multiplication (10 min)

2. Elimination w/ multiplication: which eqn to multiply & by what # (10 min)

3. Concept Summary: best time to use each method (10 min)

4. 5 examples: which method is best to use (10 min)

Warm Up:

1. Take out the systems packet

2. Do the back of this week’s agenda - week 19

HW#1 - HW#6 will be collected and graded tomorrow!

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

Two groups of students order burritos and tacos at Los Gallos. 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?

HW#3: Elimination Answers1. (5, 2)

2. (1, 6)

3. (6, 1)

4. (-3, 5)

5. (4, -1)

6. (2, 3)

7. (6, 18)

Ex. 1a: Elimination using Multiplication (easiest to eliminate the y

variables)

5x + 6y = -8

2x + 3y = -5

Ex. 1b: Elimination using Multiplication (eliminate the x variables)

5x + 6y = -8

2x + 3y = -5

Ex 2a: Elimination using Multiplication(Eliminate the x variables)

9x + 5y = 34

8x – 2y = -2

Ex 2b: Elimination using Multiplication(Eliminate the y variables)

9x + 5y = 34

8x – 2y = -2

Ex3a: Elimination using Multiplication (Eliminate the x variables)

3x + 3y = 9

4x + 2y = 8

Answer: (1, 2)

Ex3b: Elimination using Multiplication (Eliminate the y variables)

3x + 3y = 9

4x + 2y = 8

Answer: (1, 2)

HW#4: Elimination Answers1. (2, -3)

2. (1, 2)

3. c = 3.95, a = 5.95

4. a.) (4, 1) d.) (0, 3) (2, 5)

HW#5: Any Method Answers1. D

2. D

3. C

HW#6: Real-Life Examples Answers1. a.) 10 t-shirts need to be ordered for both shops to charge

an equal amount. b.) If 9 or less t-shirts are ordered, Shop A is less expensive. If 11 or more t-shirts are ordered, Shop B is less expensive

2. a.) C = 1.75 + 1.80m C = 2.50 + 1.20m

b.) At 1.25 miles the companies charge the same amount.

3. C = #childrenA = #adultsC + A = 22001.50C + 4.00A = 50501500 children and 700 adults attended.

Fill in the chart below:

Method The Best Time to Use

Graphing

Substitution

Elimination using Addition

Elimination using Subtraction

Elimination using Multiplication

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 equations.

Substitution

Elimination using Addition

Elimination using Subtraction

Elimination using Multiplication

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 equations.

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

Elimination using Addition

Elimination using Subtraction

Elimination using Multiplication

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 equations.

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

Elimination using Addition

If one of the variables has opposite coefficients.

Elimination using Subtraction

Elimination using Multiplication

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 equations.

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

Elimination using Addition

If one of the variables has opposite coefficients.

Elimination using Subtraction

If one of the variables has the same coefficients.

Elimination using Multiplication

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 equations.

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

Elimination using Addition

If one of the variables has opposite coefficients.

Elimination using Subtraction

If one of the variables has the same coefficient.

Elimination using Multiplication

If none of the coefficients are 1 and neither of the variables can be eliminated by simply adding or subtracting the equations.

Which method is best to use? Why?

1. x = 12y – 143y + 2x = -2

Substitution; one equation is solved for x

2. 20x + 3y = 20-20x + 5y = 60

Elimination using addition to eliminate x

3. y = x + 2y = -2x + 3

Substitution; both equations are solved for y

Which method is best to use? Why?4. -20x + 3y = 20

-20x + 5y = 60Elimination using subtraction to eliminate x

5. -5x – 3y = 20

-5x + 3y = 60Elimination using subtraction to eliminate x OR

elimination using addition to eliminate y

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?

HW#6, Problem #3

How a school uses systems of equations to see how many tickets they sell

Your class sells a total of 64 tickets to the school play. A student ticket costs $1 and an adult ticket costs $2.50. Your class collects $109 in total tickets sales. How many adult and student tickets did you sell?

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?

How a bakery uses systems of equations to track their inventory

La Guadalupana Bakery sells pies for $6.99 and cakes for $10.99. The total number of pies and cakes sold on a busy Friday was 36. If the amount collected for all the pies that day was $331.64, how many of each type were sold?

How a math student uses systems of equations to solve math puzzles!

The sum of two numbers is 25 and their difference is 7. Find the numbers.

How a math student uses systems of equations to solve math puzzles!

Twice one number added to another is 18. Four times the first number minus the other number is 12. Find the numbers.

If you would like additional practice to prepare for Monday’s system of equations test, on the Infinity website, you will find:

1. System of equations study guide (15 problems)

2. PPT – System practice (12 problems)