systems of equations. greenland sharkspiny dogfish shark growth rate: 0.75 cm/yr birth length: 37 cm...

Systems of Equations

Greenland Shark Spiny Dogfish Shark

growth rate: 0.75 cm/yrBirth length: 37 cm

growth rate: 1.5 cm/yrBirth length: 22 cm

At what age will they be the same length?

Use a table and a graph to figure this out. (do table on dry erase?)

Demonstrate graphs on calculator.Slide 5 has the steps.

Are there other ways?

There are 3 ways that you probably have seen for solving systems of equations.

(1) Substitution (2) Linear Combination(3) Graphing

What is this?

GraphingStep 1) Write the equations in the form y mx b

0.75 37 eq. (1)

1.5 22 eq. (2)

y x

y x

Step 2) Enter the functions into Y1 and Y2 (2 / 3) (2 / 3)x

Step 3) Press the graph button

Step 4) Press 2nd TRACE Select 5: intersect Press Enter twice Move cursor to intersection point and press Enter

GraphingWe now know what it means if there is one intersection point. What do you think it means if there are no intersection points?

How about if there are an infinite number of intersection points?

Assignment: (p138) 10-16

Where do you want to work?

$35 per day plus 10% commission on all sales

$10 per day plus 18% commission on all sales

How much would you need to sell per day to make the same amount at each store?

Piano Lessons

$6 times the cost of one lesson plus a one time fee is $300

What is the one time fee and cost of one lesson?

$12 times the cost of one lesson plus a one time fee is $480

$120 - one time fee$30 - one lesson

Slides 9-24 are examples of substitution and linear combination

(1) SubstitutionThis might be the method students are the most familiar with.

2 3 2 eq. (1)

5 2 14 eq. (2)

x y

x y

Find the values of x and y that satisfy this system of equations.

Step 1) Solve one of the equations for either x OR y. Any variable will lead to the answer. Your goal is to find the variable that is the easiest to solve for. (in this case, they are all the same level of difficulty)

(1) Substitution

2 3 2 eq. (1)

5 2 14 eq. (2)

x y

x y

Find the values of x and y that satisfy this system of equations.

Step 1) Solve one of the equations for either x OR y. Any variable will lead to the answer. Your goal is to find the variable that is the easiest to solve for.

Step 2) Substitute this value of x (eq. 3) into eq (2)

31 eq. (3)

25 2 14 eq. (2)

x y

x y

(1) SubstitutionStep 1) Solve one of the equations for either x OR y. Any variable will lead to the answer. Your goal is to find the variable that is the easiest to solve for.

Step 2) Substitute this value of x (eq. 3) into eq (2)

31 eq. (3)

25 2 14 eq. (2)

x y

x y

3 15 45 1 2 14 5 14

2 2 2

19 5 14 eq. (4)

2

y y y y

y

(1) SubstitutionStep 1)

Step 2) Substitute this value of x (eq. 3) into eq (2)

31 eq. (3)

25 2 14 eq. (2)

x y

x y

195 14 eq. (4)

2y

Step 3) Solve eq. 4 for y

19 219 19 2 2

2 19y y y

(1) SubstitutionStep 1)

Step 2) Substitute this value of x (eq. 3) into eq (2)

31 eq. (3)

25 2 14 eq. (2)

x y

x y

195 14 eq. (4)

2y

Step 3) Solve eq. 4 for y: 2y

Step 4) Substitute the value of y from step (3) into eq. (3) to get x.

3 31 (2) 1 3 1 2 2

2 2x y x

(1) Substitution

2 3 2 eq. (1)

5 2 14 eq. (2)

x y

x y

Find the values of x and y that satisfy this system of equations.

Answer: 2, 2x y

Step 5) (OPTIONAL ) Check your answer to confirm.

2(2) 3(2) 4 6 2

5(2) 2(2) 10 4

Yes,

14

this is true.

Yes, this is tru .e

(1) Substitution

4 2 10 eq. (1)

5 2 11 eq. (2)

x y

x y

Your Turn. Use the method of substitution to solve the following system for x and y.

Take a few minutes. Then compare your answer with your neighbor's.This will be graded. You will turn this in when you finish. .

Answer.

(1) Substitution

4 2 10 eq. (1)

5 2 11 eq. (2)

x y

x y

Your Turn. Use the method of substitution to solve the following system for x and y.

Answer: 1, 3x y

Take a few minutes. Then compare your answer with your neighbor's.This will be graded. You will turn this in when you finish. .

(2) Linear Combination

2 3 2 eq. (1)

5 2 14 eq. (2)

x y

x y

Find the values of x and y that satisfy this system of equations. (same system as before)

Step 1) The goal of this step is to multiply each

equation by a number that will allow you to add the equations together to eliminate a variable. This is a bit confusing trying to describe with words. Look at the following.I will try to eliminate the “y”.

2 3 2 eq. (1)

5 2 14 eq. (2)

x y

x y

Find the values of x and y that satisfy this system of equations. (same system as before)

Step 1)

2 3 2 eq. (1)

5 2 14

2

2 2

eq.3 (2 3 ) 3

xx y

xx y

4 4 eq. (1)

15 42 eq. (26

2

)

6

3

x

x

y

y

x

x

(2) Linear Combination

Step 1) 4 4 eq. (1)

15 42 eq. (26

2

)

6

3

x

x

y

y

x

x

Step 2) Add the two equations from step (1).

4 4

15 42

19 38

6

6

0

y

y

x

x

x

Step 3) Solve the result of step (2) for x.

(2) Linear Combination

Step 1) 4 4 eq. (1)

15 42 eq. (26

2

)

6

3

x

x

y

y

x

x

Step 2) Add the two equations from step (1).

19 38x

Step 3) Solve the result of step (2) for x.

2x

Step 4) Substitute x into either eq. (1) or eq. (2) to find the value of y. 2 3 2 eq. (1)

5 2 14 eq. (2)

x y

x y

(2) Linear Combination

Step 1) 4 4 eq. (1)

15 42 eq. (26

2

)

6

3

x

x

y

y

x

x

Step 2) Add the two equations from step (1). 19 38x Step 3) Solve the result of step (2) for x. 2x Step 4) Substitute x into either eq. (1) or eq. (2) to find the value of y.

2 3 2 eq. (1)

5 2 14 eq. (2)

x y

x y

2(2) 3 2 3 6 2y y y

(2) Linear Combination

2 3 2 eq. (1)

5 2 14 eq. (2)

x y

x y

Find the values of x and y that satisfy this system of equations.

Answer: 2, 2x y This is the same as before, which is good since we started with the same system of equations.

Step 5) (OPTIONAL ) Check your answer to confirm. This is exactly like we did before.

(2) Linear Combination

19 3 21 eq. (1)

6 42 eq. (2)

x y

x y

Your Turn. Use the method of linear combination to solve the following system for x and y.

Answer

Take a few minutes. Then compare your answer with your neighbor's.This will be graded. You will turn this in when you finish. .

(2) Linear Combination

19 3 21 eq. (1)

6 42 eq. (2)

x y

x y

Your Turn. Use the method of linear combination to solve the following system for x and y.

Answer: 0, 7x y

Take a few minutes. Then compare your answer with your neighbor's.This will be graded. You will turn this in when you finish. .

(2) Linear Combination

Assignment

Assignments:

1) (p146) 16-21, 29-322) (p147) 44-543) (p262-263) 8,10,20,23-25, 31-34 (no calculator on 31-34),47