manual62.doc · web viewx + y = 12 3x + 2y = 26 3. x = 4y – 5 (-9, -1) 4. y = 3x + 2 (3, 11) 2x +...

Unit 6 – Systems Length of section

6-1 Graphing Systems of Equations 3 days

6-2 Substitution 3 days

6-3 Adding and Subtracting 3 days

6.1 - 6.3 Quiz 1 days

6-4 Multiplication 4 days

6-5 Graphing Systems of Inequalities 3 days

Test Review 1 day

Test 1 day

Cumulative Review 1 day

Total days in Unit 6 – Systems = 20 days

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Review QuestionWhat makes an equation linear? Exponents on variables are 1What is a solution to a linear equation? Point (2, 7). Notice this point “works” in y = 3x +1.

DiscussionWhat do you think a system of equations is? 2 or more equationsWhat is a solution to a system of equations? Point that works in all equations

y = 2x – 3 y = -3x + 7

Notice that (2, 1) works in both equations. What would that look like? Two lines(2, 1) is where the two lines would intersect.

How many ways can two lines intersect?

# of Intersections # of Solutions How?

1 1 Different m’s

0 0 Same m’s;Different intercepts

Infinite Infinite Same m’s;Different intercepts

SWBAT find the solution to a system of equations by graphing

Example 1: Graph each line to find the solution.y = 2x – 3 y = -3x + 7

(2, 1) is the solution because that is the intersection point.How do you know that your answer is correct? (2, 1) “works” in both equations

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Section 6-1: Graphing Systems of Equations (Day 1) (CCSS: N.Q.3, A.CED.3, A.REI.6, A.REI.7, A.REI.11)

Example 2: Graph each line to find the solution.y = 4x – 3 y = -3x + 7

The solution looks like it would be (2, 2).How do you know that your answer is incorrect? (2, 2) doesn’t “work” in either equationThe correct solution is (10/7, 19/7).

Hmmmm?!?What issue do you see with graphing to find the solution? It is not exact.

Example 3: Graph each line to find the solution.y = 2x – 3 y = 2x + 3

The solution is No Solution.What does the answer of No Solution mean? No points will work in both equations.

239

Example 4: Graph each line to find the solution.y = 2x + 12y – 4x = 2The solution is Infinite Solutions.

What does the answer of infinite solutions mean? There are an infinite amount of points that will work in both equations.Can someone give me one of the possible answers? (0, 1) (1, 3)…

You Try!Graph each line to estimate the solution.

1. y = 2x + 3 (.5, .5) 2. y = 4x – 1 No Solution y = -3x + 1 y – 4x = 2

3. y = 4x + 1 Infinite Solutions 4. y + 3x = 1 (1, -3) 3y = 12x + 3

________ 5. 3y – 8 = 4x (2, 5) 6. y = 5 No Solution x = 2 y = -2

What did we learn today?

Graph each line to estimate the solution.

1. y = 3x + 3 2. y = -3x + 4 y = -2x + 2 y = -3x + 2

240

Section 6-1 Homework (Day 1)

3. y = 4 4. y + 3x = 1 y = 6

________

5. y = -2x – 1 6. y – 4 = 2x x = 1 y = 2x + 6

7. y – 5x = 2 8. 4y = 3x – 2 y = 5x + 2 y = -2x – 2

9. y = 3x + 1 10. x = -3 4y = 12x + 4 x = 3

241

Review QuestionWhat does a solution to a system of equations look like? Use your arms as lines to demonstrate each possibility. Point, Infinite, No Solution

DiscussionCan you look at a system of equations and tell whether it will have 1, infinite, or no solution? How? Yes, look at the slopes. Different Slopes: 1 solution, same slopes: no solution, same equation: infinite

SWBAT find the solution to a system of equations by graphing

Example 1: How many solutions? Estimate the solution by graphing.

y = 2x + 3y = 2x + 5No Solution; Same slopes


y = -3x + 2y = 2x + 51; Different slopes, (-1, 3)

242

Section 6-1: Graphing Systems of Equations (Day 2) (CCSS: N.Q.3, A.CED.3,

A.REI.6, A.REI.7, A.REI.11)


y = 2x + 54y – 8x = 20Infinite Solutions; Same equations

You Try!How many solutions? Then estimate the solution by graphing.1. y = -4x + 1 1; Different slopes 2. 3y = 2x + 3 No Solution; Same slopes y = 2x – 3 (1, -2)

_________

3. x + 2y = 3 1; Different slopes 4. y = 2x – 3 Infinite Solutions; Same equation 3x – y = -5 (-3, -3) 4x = 2y + 6


State how many solutions there are going to be then graph each line to find the solution.

1. y = 4x + 1 2. y = -2x + 5 y = -2x + 1 y + 2x = 2

3. x = 2 4. y + 2x = 1 x = 3

________

243


5. 2y + 4x = -2 6. y = 3x – 1 y = -2x – 1 -3x = y + 2

7. y – 2x = 2 8. 3y = 4x – 2 y = 2x + 2 y = -2x – 2

9. y = x + 1 10. y = 4 2y = -2x + 2 x = -1

244

Review QuestionWhat are the three possibilities for a solution to a system of equations? Use your arms as lines to demonstrate each possibility. Point, Infinite, No Solution

DiscussionWhat is the major issue with solving a system of equations by graphing? It is not precise.Today, we will be using the graphing calculator to find exact solutions.

SWBAT find the solution to a system of equations by using a graphing calculator

Example 1: How many solutions? 1 y = 4x – 2y = -2x + 3Graph to find the solution. (1, 1)

How do you know that your answer is wrong? That point does not work in both equations.Let’s find the exact answer using the graphing calculator. 1. Press the “y =” button. Enter each equation.2. Press graph.3. Press 2nd, then trace.4. Scroll down to 5: Intersect. Press enter.5. Press enter 3 times. (.83, 1.33)

Example 2: How many solutions? 0; Empty Sety = -2x + 3y + 2x = -5Let’s confirm our answer using the graphing calculator.(See instructions from above.)

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Section 6-1: Graphing Systems of Equations (Day 3) (CCSS: N.Q.3, A.CED.3, A.REI.6, A.REI.7, A.REI.11)

Example 3: How many solutions? Infinite Solutionsy = 4x + 52y – 8x = 10Let’s confirm our answer using the graphing calculator.(See instructions from above.)

Example 4: How many solutions? 1y = 8x – 13y + 5x = 55Let’s find the exact answer using the graphing calculator.(See instructions from above.)

Why can’t we see the intersection point? We have to change the window.The solution is (2, 15).


Estimate the answer by graphing. Then find the exact answer using the graphing calculator.

1. y = 4x – 3 y = -2x + 2

3. 5y – 4x = 5 y = -2x + 2

2. y = 2x + 6 y – 2x = 1

4. 3y = 6x + 9 y = 2x + 3

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Section 6-1 In-Class Assignment (Day 3)

Use the graphing calculator to find the exact answer. Then sketch the graph.

5. y = -3x + 2 y = 2x – 1

7. y = 5x – 1 y = 5x + 2

9. y – 4x = 3 y = 4x + 3

6. 5y = 3x – 2 y = -5x – 2

8. y = x + 1 3y = 3x + 3

10.

247

Review QuestionWhat issue do we have with graphing? It isn’t exact.Today we will discuss a way to find the exact answer to a system of equations.

DiscussionSolve: 2x + 5 = 11. How can you check to make sure that ‘3’ is the correct answer? Substitute ‘3’ in for x.What does substitution mean? Replacing something with something else.

That is what we will be doing today. This allows us to find exact answers to systems of equations. Since graphing did not.

Solving 2x + 5 = 11 is pretty easy. Why would solving the following system be difficult?y = 3x + 52x + 4y = 8

There are two equations and two variables. If we could get it down to one equation/one variable, it would be easy. This is what substitution allows us to do.

SWBAT find the solution to a system of equations by using substitution

Example 1: y = 3x – 2 2x + 3y = 27

We need to get rid of one variable/equation. We do this by substitution.What is ‘y’ equal to? y = 3x – 2

So we substitute 3x – 2 in for ‘y’ in the second equation. When we do this, the second equation becomes:2x + 3(3x – 2) = 27. Now solve. Notice how substitution got rid of an equation and a variable.2x + 9x – 6 = 2711x – 6 = 2711x = 33 x = 3

Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.y = 3(3) – 2 y = 9 – 2y = 7The final answer is (3, 7).What does the answer (3, 7) mean? That is the point of intersection.

Section 6-2: Substitution (Day 1) (CCSS: A.CED.3, A.REI.5, A.REI.7)

Example 2: y = 3x + 5 6x – 4y = -32

We need to get rid of one variable/equation. We do this by substitution.What is ‘y’ equal to? y = 3x + 5

So we substitute 3x + 5 in for ‘y’ in the second equation. When we do this, the second equation becomes:6x – 4(3x + 5) = -32. Now solve. Notice how substitution got rid of an equation and a variable.6x – 12x – 20 = -32-6x – 20 = -32-6x = -12 x = 2

Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.y = 3(2) + 5 y = 6 + 5y = 11The final answer is (2, 11).What does the answer (2, 11) mean? That is the point of intersection.

Example 3: x + 5y = -3 3x – 2y = 8

What is different about this problem? None of the variables are solved for alreadyWhat variable should we solve for? Why? The ‘x’ in the first equation. It is the easiest.When we solve for the ‘x’ in the first equation, we get: x = -3 – 5y.

So we substitute -3 – 5y in for ‘x’ in the second equation. When we do this, the second equation becomes: 3(-3 – 5y) – 2y = 8. Now solve. Notice how substitution got rid of an equation and a variable.-9 – 15y – 2y = 8-9 – 17y = 8-17y =17 y = -1

Now substitute the ‘y’ back into the equation where we already solved for ‘x’ to get the ‘x’ value.x = -3 – 5(-1) x = -3 + 5x = 2The final answer is (2, -1).What does the answer (2, -1) mean? That is the point of intersection.

SummarizeWhen is it easy to use substitution? When a variable is solved for or can be easily solved for

You Try!1. y = 4x + 1 (3, 13) 3x + 2y = 35

2. x = 3y – 4 (2, 2) 2x + 4y = 12

3. 8x – 2y = 2 (1, 3) 3x + y = 6

4. 2x – y = -4 (13, 30) -3x + y = -9


Solve each system of equations using substitution. Confirm your answer by graphing.


1. y = 5x (2, 10) 2. y = 3x + 4 (2, 10) x + y = 12 3x + 2y = 26

3. x = 4y – 5 (-9, -1) 4. y = 3x + 2 (3, 11) 2x + 3y = -21 2x + y = 17

Solve each system of equations using substitution.

5. y = 5x + 1 (-2, -9) 6. y = 2x + 2 (3, 8) 3x + y = -15 2x – 4y = -26

7. 3x + 2y = 7 (1, 2) 8. y = 4x – 2 (2, 6) x + 3y = 7 x – 3y = -16

9. 4x + y = 16 (3, 4) 10. y = 5x (1, 5) 2x + 3y = 18 y = 3x + 2

Review QuestionWhat are the three possibilities for a solution to a system of equations? Use your arms as lines to demonstrate each possibility. Point, Infinite, No Solution

Yesterday, we just addressed the first case.

DiscussionSolve: 2x + 5 = 2x + 7. What does 5 = 7 mean? There is no solution to this problem.

Solve: 2x + 7 = 2x + 7. What does 7 = 7 mean? There are infinite solutions to this problem.


Example 1: y = 2x + 3 3x + 3y = 45


So we substitute 2x + 3 in for ‘y’ in the second equation. When we do this, the second equation becomes:3x + 3(2x + 3) = 45. Now solve. Notice how substitution got rid of an equation and a variable.3x + 6x + 9 = 459x + 9 = 459x = 36 x = 4



Example 2: y = 2x + 3 -4x + 2y = 6


So we substitute 2x + 3 in for ‘y’ in the second equation. When we do this, the second equation becomes:-4x + 2(2x + 3) = 6. Now solve. Notice how substitution got rid of an equation and a variable.-4x + 4x + 6 = 6 6 = 6

When does 6 = 6? AlwaysWhat does that mean? We have infinite solutionsWhat kind of lines do we have? They are the same line. (On top of each other.)

Example 3: -x + y = 4 -3x + 3y = 10

We need to get rid of one variable/equation. We do this by substitution.What variable should we solve for? Why? ‘y’ in the first equation. It is the easiest.When we solve for the ‘y’ in the first equation, we get: y = 4 + x.

So we substitute 4 + x in for ‘y’ in the second equation. When we do this, the second equation becomes:-3x + 3(4 + x) = 10. Now solve. Notice how substitution got rid of an equation and a variable.-3x + 12+ 3x = 10 12 = 10

When does 12 = 10? NeverWhat does that mean? There is no solution.What kind of lines do we have? Parallel

SummarizeWhen is it easy to use substitution? When a variable is solved for or can easily be solved for

You Try!1. y = 3x + 5 (3, 14) 4x + 2y = 40

2. y = 2x + 3 No Solution -4x + 2y = 12

3. y – 3x = 4 Infinite Solutions -9x + 3y = 12

4. 4x + y = 11 (2, 3) 3x – 2y = 0



1. y = 3x (3, 9) 2. y = 4x + 3 (2, 11) 2x + 3y = 33 3x + 2y = 28


3. y = 3x + 2 No Solution 4. 4x + y = 13 (3, 1) -3x + y = 10 3x + 5y = 14

5. y = 5x + 2 Infinite Solutions 6. 3x – 2y = 4 (2, 1) -10x + 2y = 4 -4x + y = -7

7. y = 4x – 3 (4, 13) 8. 2x + y = 10 Infinite Solutions 3x + 3y = 51 6x + 3y = 30

9. y = 3x + 1 (3, 10) 10. -2x + 8y = 8 No Solution 2x + 3y = 36 x – 4y = 10


Review QuestionWhen is it easy to use substitution? When a variable is solved for or can be easily solved for

DiscussionHow do you get better at something? PracticeTherefore, we are going to practice solving systems of equations today.

We are going to have many days like this during the school year. In order for you to be successful, you need to take advantage of the time and ask questions from your classmates and teachers.


Example 1: Let’s make sure we know how to use substitution.y = 2x + 32x + 3y = 25We need to get rid of one variable/equation. We do this by substitution.What is ‘y’ equal to? y = 2x + 3

So we substitute 2x + 3 in for ‘y’ in the second equation. When we do this, the second equation becomes:2x + 3(2x + 3) = 25. Now solve. Notice how substitution got rid of an equation and a variable.2x + 6x + 9 = 258x + 9 = 258x = 16 x = 2


Let’s graph to confirm our answer.


You Try!1. y = 5 – 2x (1, 3) 2. y = 4 – 2x (1, 2) 3y + 3x = 12 2x – y = 0

3. x + y = 6 No Solution 4. 2x + y = 3 Infinite Solutions 3x + 3y = 3 4x + 2y = 6



1. y = 3x (-1, -3) 2. y = 3 (1, 3) 2x + 3y = -11 3x + 2y = 9

3. x = -4 (-4, 2) 4. 2x + y = 5 No Solution 3x + y = -10 4x + 2y = 2

5. y = 2x + 1 (1, 3) 6. 6x – 2y = 5 Infinite Solutions 4x + 2y = 10 -12x + 4y = -10


7. y = 3x (-3, -9) 8. x + 5y = 11 (1, 2) x + 2y = -21 3x – 2y = -1

9. y = 3x + 4 (2, 10) 10. -2x + 2y = 4 (1, 3) 2x + 3y = 34 x – 4y = -11

11. y = 4x – 6 (3, 6) 12. 2x + y = 7 (3, 1) 3x + 4y = 33 3x – 2y = 7

13. y = 3x + 1 (-2, -5) 14. x + 3y = 14 (5, 3) 2x + 3y = -19 2x – 4y = -2

15. x = 2y + 4 (8, 2) 16. -3x + 2y = -8 (2, -1) 2x + 3y = 22 x – 4y = 6


Review QuestionWhat are the three possibilities for a solution to a system of equations? Use your arms as lines to demonstrate each possibility. Point Infinite No Solution DiscussionWhen is it easy to use substitution? When a variable is solved for or can be easily solved forWhy do we substitute something in for a variable? It allows us to get rid of one variable/equation.How does this help us? We can solve one equation with one variable

Why wouldn’t substitution be good for the following system? When you solve for one of the4x + 5y = 12 variables, the result will be a fraction4x – 3y = -4

What is something else that we could do? Subtract; it would get rid of one variable/equation.(Remember our goal is to get rid of one variable/equation)Remember the section title;Remember the Alamo!)

Why are we allowed to add or subtract two equations to each other? Since both sides are equal to each other, we can add/subtract to both sides.Just like: 2x + 5 = 11 - 5 - 5

So, when is it good to use addition/subtraction? When the coefficients are the sameHow do you know whether to add or subtract? Same signs: subtract; Different signs: add

SWBAT find the solution to a system of equations by using addition/subtraction

Example 1: 3x – 2y = 4 Add the second equation to the first one. 4x + 2y = 10

7x = 14 x = 2

Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.3(2) – 2y = 4 6 – 2y = 4-2y = -2y = 1The final answer is (2, 1).What does the answer (2, 1) mean? That is the point of intersection.Why would we use addition not subtraction? Because it eliminates the y’s

Section 6-3: Adding/Subtracting (Day 1) (CCSS: A.CED.3, A.REI.7)

Example 2: 4x + 5y = 12 Subtract the second equation from the first one. 4x – 3y = -4

8y = 16 y = 2

Now substitute the ‘y’ back into either of the first two equations to get the ‘x’ value.4x + 5(2) = 124x + 10 = 124x = 2 x = .5The final answer is (.5, 2).What does the answer (.5, 2) mean? That is the point of intersection.Why would we use subtraction not addition? Because it eliminates the x’s

Example 3: 2x – 3y = 10 (-1, -4) 2x = y + 2What is different about this system? The x’s and y’s are not on the same side of the equation.After a little bit of Algebra, we get the following system:2x – 3y = 10 Subtract the second equation from the first one. (-1, -4) 2x – y = 2 -2y = 9 y = -4

Now substitute the ‘y’ back into either of the first two equations to get the ‘x’ value.2x – 3(-4) = 10 2x + 12 = 102x = -2x = -1The final answer is (-1, -4).What does the answer (-4, -1) mean? That is the point of intersection.Why would we use subtraction not addition? Because it eliminates the x’s

SummarizeWhen is it easy to use addition/subtraction? When the coefficients are the sameHow do you know whether to add or subtract? Same signs: subtract; Different signs: add

You Try!1. 4x – 5y = 10 (5, 2) 2. 3x + 5y = -16 (-2, -2) 2x + 5y = 20 3x – 2y = -2 3. y = 4x + 2 (1, 6) 4. -6x + 2y = 2 (0, 1) 3x + 4y = 27 6x = 3y – 3

5. 4x + 2y = 16 No Solution 6. 3x + y = 16 (5, 1) 4x + 2y = 10 6x – 3y = 27


Use addition, subtraction, or substitution to solve each of the following systems of equations.

1. 3x + 2y = 22 (6, 2) 2. 3x + 2y = 30 (4, 9) 3x – 2y = 14 y = 2x + 1

3. 3x – 5y = -35 (-5, 4) 4. 5x + 2y = 12 (2, 1) 2x – 5y = -30 -5x + 4y = -6

5. 4x = 7 – 5y (.5, 1) 6. x = 6y + 11 (23, 2) 8x = 9 – 5y 2x + 3y = 52

7. x – 3y = 7 (4, -1) 8. 3x + 5y = 12 Infinite Solutions x + 2y = 2 3x + 5y = 12

9. 4x + y = 12 (2, 4) 10. 2x + 3y = 5 (4, -1) 3x + 3y = 18 5x + 4y = 16


Review QuestionWhen is it easy to use substitution? When a variable is solved for or can be easily solve forWhen is it easy to use addition/subtraction? When the coefficients are the same DiscussionWhat method should we use for problem #10 on the homework? SubstitutionWhy does this stink? It involves fractions.

How do you get better at something? PracticeTherefore, we are going to practice solving systems of equations today.

We are going to have many days like this during the school year. In order for you to be successful, you need to take advantage of the time and ask questions from your classmates and teachers.

SWBAT find the solution to a system of equations by using addition/subtraction

Example 1: 5x – 4y = 8 Add the second equation to the first one. 4x + 4y = 28

9x = 36 x = 4

Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.5(4) – 4y = 8 20 – 4y = 8-4y = -12y = 3The final answer is (4, 3).What does the answer (4, 3) mean? That is the point of intersection.Why would we use addition not subtraction? Because it eliminates the y’s

Example 2: 5x + 5y = -5 5x – 3y = 11 8y = -16

y = -2

Now substitute the ‘y’ back into either of the first two equations to get the ‘x’ value.5x + 5(-2) = -5 5x – 10 = -55x = -5x = -1The final answer is (-1, -2).What does the answer (-1, -2) mean? That is the point of intersection.Why would we use addition not subtraction? Because it eliminates the x’s


You Try!1. 4x – 7y = -13 (2, 3) 2x + 7y = 25

2. 3x + 4y = -9 (1, -3) 3x = 2y + 9

3. y = -2x – 3 Infinite Solutions 4x + 2y = -6

4. 3x + 2y = 11 No Solution 3x + 2y = 8


Use addition, subtraction, or substitution to solve each of the following systems of equations.

1. 5x + 4y = 14 (2, 1) 2. 3x + 6y = 21 (1, 3) 5x + 2y = 12 -3x+ 4y = 9

3. 5x + 2y = 6 (4, -7) 4. y = -3x + 2 (0, 2) 9x + 2y = 22 3x + 2y = 4

5. 2x – 3y = -11 (-1, 3) 6. 6x + 5y = 8 No Solution x + 3y = 8 6x + 5y = - 2

7. x = 3y + 7 Infinite Solutions 8. 3x – 4y = -5 (1, 2) 3x – 9y = 21 3x = -2y + 7

9. 2x + 3y = 1 (-1, 1) 10. 4x – 5y = 2 (3, 2) x + 5y = 4 6x + 5y = 28


Review QuestionWhen is it easy to use substitution? When a variable is solved for or can be easily solved forWhen is it easy to use addition/subtraction? When the coefficients are the same DiscussionWhat is our goal when we are trying to solve a system of equations? Get rid of one variableHow does this help us? We can solve an equation with one variable.

SWBAT solve a word problem that involves a system of equations

Example 1: Find two numbers whose sum is 64 and difference is 42.

x + y = 64 Add the second equation to the first one.x – y = 422x = 106x = 53

Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.53 + y = 64 y = 11The two numbers are 53 and 11.

Example 2: Cable costs $50 for installation and $100/month. Satellite costs $200 for installation and $70/month. What month will the cost be the same?

C = 50 + 100m Subtract the second equation from the first one.C = 200 + 70m0 = -150 + 30m150 = 30m5 = m

What does 5 months represent? The month where it costs the same for both gyms.How could this help you decide on which company to go with? Depending on how long you are going to keep your cable.



Use addition, subtraction, or substitution to solve each of the following systems of equations. Graph to confirm your answer in problem #1.

1. 2x + 2y = -2 (2, -3) 2. 4x – 2y = -1 (-1, -1.5) 3x – 2y = 12 -4x + 4y = -2

3. 6x + 5y = 4 (-1, 2) 4. x = 3y + 7 (4, -1) 6x – 7y = - 20 3x + 4y = 8

5. 2x – 3y = 12 (6, 0) 6. 3x + 2y = 10 No Solution 4x + 3y = 24 3x + 2y = -8

7. -4x – 2y = -10 Infinite Solutions 8. 8x + y = 10 (1, 2) 2x + y = 5 2x – 5y = -8

Write a system of equations. Then solve.9. The sum of two numbers is 70 and their difference is 24. Find the two numbers. 23, 47

10. Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers. 5, 8

11. Two angles are supplementary. The measure of one angle is 10 more than three times the other. Find the measure of each angle. 42.5, 137.5

12. Johnny is older than Jimmy. The difference of their ages is 12 and the sum of their ages is 50. Find the age of each person. 31, 19

13. The sum of the digits of a two digit number is 12. The difference of the digits is 2. Find the number if the units digit is larger than the tens digit. 5 and 7

14. A store sells Cd’s and Dvd’s. The Cd’s cost $4 and the Dvd’s cost $7. The store sold a total of 272 items and took in $1694. How many of each was sold?

202, 70


Review QuestionWhen is it easy to use substitution? When a variable is solved for or can be easily solved forWhen is it easy to use addition/subtraction? When the coefficients are the same DiscussionWhat is our goal when we are trying to solve a system of equations? Get rid of one variableHow does this help us? We can solve an equation with one variable.What method would you use to solve the following system of equations?

2x + 3y = 56x + 4y = 16

Why wouldn’t substitution be good? It would involve fractions.Why wouldn’t add/subtract be good? It will not eliminate any of the variablesWe need something else.

What could we do to the first equation to make it so we could subtract? Multiply by 3Why are we allowed to do this? You are allowed to multiply the entire equation by whatever you want.

SWBAT solve a system of equations using multiplication

Example 1: 9x + 8y = 10 Multitpy by 2, then subtract 18x + 16y = 20 18x + 3y = 33 18x + 3y = 33

13y = -13y = -1

Now substitute the ‘y’ back into either of the first two equations to get the ‘x’ value.9x + 8(-1) = 10 9x – 8 = 109x = 18x = 2The final answer is (2, -1).

Example 2: 2x + 3y = 5 5x + 4y = 16

How is this problem different from the previous one? You need to multiply both equations by something to get the coefficients the same. Then you can subtract.

Multiply the 1st equation by 5: 10x + 15y = 25Multiply the 2nd equation by 2: 10x + 8y = 32

7y = -7 y = -1

Section 6-4: Multiplication (Day 1) (CCSS: A.CED.3, A.REI.5, A.REI.7)


Hmmmm?!?When should we use multiplication? When the coefficients are differentHow about division? It is the same as multiplying.Dividing by 2 is the same as multiplying by 1/2

You Try!1. 2x + 3y = 8 (1, 2) 2. 4x + 5y = -7 (2, - 3) 4x + 5y = 14 6x – 3y = 21

3. x = 5y + 7 (-18, -5) 4. 3x – 4y = 12 No Solution 2y – x = 8 3x – 4y = -14

5. -x + y = -15 (13, -2) 6. 2x – 3y = 1 (2, 1) -4y = x – 5 5x + 5y = 15 What did we learn today?

Use addition, subtraction, substitution, or multiplication to solve each of the following systems of equations.

1. 5x + 4y = 19 (3, 1) 2. 2x + 6y = 28 (2, 4) 2x + 2y = 8 3x + 4y = 22

3. 5x + 2y = 4 (2, -3) 4. y = 4x + 3 (1, 7) 8x + 2y = 10 4x + 2y = 18

5. 2x – 3y = -16 (-2, 4) 6. 3x + 2y = -11 (-3, -1) 3x + 3y = 6 6x + 5y = -23


7. 4x = 4y – 4 (1, 2) 8. 3x – 4y = 10 Infinite Solutions 3x – 9y = -15 9x – 12y = 30

9. 2x + 3y = 1 (-1, 1) 10. 2x – 5y = -2 (4, 2) x + 5y = 4 6x + 5y = 34

11. x = 2y + 3 (11, 4) 12. 3x – 4y = 10 No Solution 3x + 2y = 41 3x = 4y + 5

13. 2x + 3y = -1 (-2, 1) 14. 3x – 2y = 7 (3, 1) 2x + 5y = 1 5x + 3y = 18

Review QuestionWhen do we use multiplication to solve a system of equations? When the coefficients are different

Why is it important to know all of the different methods? Makes it easier; must know + and – to use multiplication DiscussionWhich method should you use?1. 4x + 6y = 12 3x – 2y = 13Multiplication

2. y = 3x + 2 2x – 5y = 12Substitution

3. 2x + 5y = -11 -2x – 2y = 11 Addition

SWBAT solve a system of equations using multiplication

Example 1: 2x + 3y = 5 -5x – 2y = -18

Multiply the 1st equation by 5: 10x + 15y = 25Multiply the 2nd equation by 2: -10x – 4y = -36

11y = -11 y = -1


You Try!1. 2x + 4y = 10 (3, 1) 3x – 2y = 7

2. y = 3x + 2 (-2, -4) 4x – 5y = 12

Section 6-4: Multiplication (Day 2) (CCSS: A.CED.3,

3. 2x + 5y = -6 (-3, 0) -5x – 3y = 15

4. 4x + 3y = 15 (3, 1) 2x – 3y = 3 What did we learn today?


1. x = 5y – 6 (4, 2) 2. -2x + y = 5 (-3/2, 2) x + 2y = 8 2x + 3y = 3

3. 2x + 3y = 6 No Solution 4. 3x + 2y = 7 (1, 2) 4x + 6y = 18 4x + 7y = 18

5. 3x – 2y = -7 (1, 5) 6. 3x = 2 – 7y Infinite Solutions y = x + 4 14y = -6x + 4

7. 4x + 6y = -10 (-1, -1) 8. 8x – 7y = 5 (-2, -3) 8x – 3y = -5 3x – 5y = 9

9. 6x + 3y = -9 (-2, 1) 10. 2x = 2y + 6 (4, 1) 2x – 3y = -7 5x – 2y = 18


Review QuestionWhen do we use multiplication to solve a system of equations? When the coefficients are different

Why is it important to know all of the different methods? Makes it easier; must know + and – to use multiplication DiscussionToday we are going to solve some word problems that require multiplication to solve. We solved some word problems that required adding and subtracting.

What is difficult about these problems? Setting up the initial system

SWBAT solve a word problem that involves multiplication to solve

Example 1: Johnny has $2.55 in nickels and dimes. He has a total of 31 coins. How many of each coin does he have?

.05n + .10d = 2.55n + d = 31

Multiply the 1st equation by 100: 5n + 10d = 255Multiply the 2nd equation by 5: 5n + 5d = 155

5d = 100 d = 20

Now substitute the ‘d’ back into the second equation to get the ‘n’ value.n + 20 = 31 n = 11The final answer is 20 dimes and 11 nickels.

Example 2: It costs $8 for adults and $5 for kids at the movie theatre. The theatre sold 107 tickets and collected a total of $670. How many of each ticket did they sell?

8a + 5k = 670a + k = 107

Leave the 1st equation alone: 8a + 5k = 670Multiply the 2nd equation by 5: 5a + 5k = 535

3a = 135 a = 45

Now substitute the ‘a’ back into the second equation to get the ‘k’ value.45 + k = 107 k = 62The final answer is 45 adults and 62 kids.

Section 6-4: Multiplication (Day 3) (CCSS: A.CED.3, A.REI.5, A.REI.7)



1. y = 3x – 2 (3, 7) 2. 4x + 6y = 0 (-3, 2) x + 2y = 17 4x + 3y = -6

3. 4x + 5y = 6 (-1, 2) 4. y = 4x – 3 (1, 1) 6x – 7y = -20 2x – y = 1

5. 2x – 5y = -2 (4, 2) 6. 2x – 4y = 8 No Solution 4x + 5y = 26 x – 2y = 3

Write a system of equations. Then solve.

7. Timmy made 145 baskets this year. Some were 2 pointers, some were 3 pointers. He scored a total of 335 points. How many 2 and 3 pointers did he make? 2 pointers: 100, 3 pointers: 45

8. Amy is 5 years older than Ben. Three times Amy’s age added to six times Ben’s age is 42. How old are Amy and Ben? Amy: 8, Ben: 3

9. The school cafeteria sold a total of 140 lunches. Some of the lunches were pizza and some were spaghetti. Pizza costs $1.50 and spaghetti costs $2. If the cafeteria collected $239, how many of each lunch did they sell? Pizza: 82, Spaghetti: 58

10. Two numbers add up to 82. Three times the bigger number minus two times the smaller number is 131. What are the two numbers? 59, 23


Review QuestionWhen is it easy to use substitution? When a variable is solved for or can be easily solved forWhen is it easy to use addition/subtraction? When the coefficients are the sameWhen is it easy to use multiplication? When the coefficients are different DiscussionIf you truly understand something, then you can talk freely about it. Specifically, you should be able to come up with your own explanations about the topic. This is what we will be doing today.

SWBAT make up a word problem that requires a system of equations to solve

You are going to make up your own problems today. In order to make up your own problems, you will have to work backwards in order to ensure your answer will make sense.

The first type of problem that you will make up will involve buying two different things. First, figure out what the two things are going to be. Next, make up how many of each thing you are going to buy. Finally, make up how much each thing costs. This will ensure that your system “works out”.

“Two Things”- Just ThinkingJimmy bought 2 things (shirt, pants).I’m thinking 8 shirts, 4 pants.The shirts are $12. The pants are $20.Therefore, he bought a total of 12 items for a total cost of $176. ($96 shirts, $80 pants)This will lead us to our actual problem…

“Two Things”- Actual ProblemJimmy bought some shirts @ $12 each. He bought some pants @ $20 each. He bought a total of 12 items. He spent a total $176. How many of each did he buy?12s + 20p = 176 s + p = 12 s = 8, p = 4

The second type of problem that you will make up will involve two different numbers. First, figure out what the two numbers are going to be. Next, figure out two different ways the numbers are related.

“Two #’s”- Just ThinkingThe two numbers I am thinking of are 12 and 26. Therefore, my problem would be: The two numbers add up to 38. If you double the first number then add two you will get the second number. This will lead us to our actual problem…

Section 6-4: Multiplication (Day 4) (CCSS: A.CED.3,

“Two #’s”- Actual ProblemTwo numbers add up to 38. If you double the first number then add two you will get the second number. What are the two numbers?x + y = 382x + 2 = yx = 12, y = 26

ActivityMake up and solve two word problems. The first problem will be “two things” and the second problem will be “two numbers”.

For each problem:1. Write a paragraph explaining the problem.2. Write an appropriate system of equations.3. Write a complete solution.

The problems should have solutions that “work out” nicely and make sense. We shouldn’t have a problem where we went to the mall and bought 128 pairs of shoes for $3 each.

* You can use HW problems to help you.


Review QuestionWhen is it easy to use substitution? When a variable is solved for or can be easily solved forWhen is it easy to use addition/subtraction? When the coefficients are the sameWhen is it easy to use multiplication? When the coefficients are different DiscussionHow do you find the solution to a system of equations by graphing? Find the intersection point.What does that point represent? The point that will “work” in both equations.How do you find the solution to a system of inequalities by graphing?Let’s come back to that in a minute…

What did the graph of y > 3x + 1 look like? Line with a shaded region.

What does the solution look like? The shaded region.What do you think the solution to a system of inequalities looks like? Intersection of shaded regions.

SWBAT graph a system of inequalities to find the solution set

Example 1: Graph: y < -2x + 1

What does the answer mean? Any point in the shaded region will “work.”

Section 6-5: Graphing Systems of Inequalities (Day 1) (CCSS: A.CED.3, A.REI.7)

Example 2: Graph: 2x + y > 4 y < -2x – 1

Is it possible to have parallel lines and the answer not be empty set? How? Yes, if and only if their shaded regions intersect

Example 3: Graph: y < 4 -3y < 3x + 6

You Try!1. y < -4x + 1 Start at (0, 1), down 4 over 1 y > 2x – 4 Start at (0, -4), up 2 over 1

2. y + 3x > 2 Start at (0, 2), down 3 over 1 y < 3 Horizontal line at 3

3. y > -4 Horizontal line at -4 x < 3 Vertical line at 3

4. y < x – 1 Start at (0, -1), up 1 over 1 -2y < -4x – 2 Start at (0, 1), up 2 over 1

How do you know that the lines aren’t parallel? Different slopes


Solve each system of inequalities by graphing.

1. y > 4x + 1 2. y < -3x + 5 y < -2x + 1 y < -3x + 1

3. y > 4 4. y + 3x < 1 y > 6

________

5. y > -2x – 1 6. y – 4 < 2x x > 1 y > 2x + 6

7. y – 3x > 2 8. 4y > 3x – 2 y < 5x + 2 y > -2x – 2

9. y > 3x + 1 10. x > -3 4y < 12x + 4 x < 3


Review QuestionHow do we know what the answer to a system of inequalities is?It is the intersecting region of the inequalities when graphed.What does this region represent? Any point in this region will “work”. DiscussionYesterday, we graphed a system of inequalities. Today, I am going to give you a graph of a system of inequalities and see if you can write the actual system. For example, what system of inequalities is represented by the graph below?

y > -2x > -3

SWBAT write a system of inequalities based on a graph

Example 1: What system of inequalities is represented by the graph below?

y < 1/2x + 1 y > -1/2x – 1 Example 2: What system of inequalities is represented by the graph below?

y < 1x + 2 y > 1x – 2



Write a system of inequalities based on the graph.1. 2.

3. 4.


5. y > 3x + 3 6. y > -2x y < -4x + 4 y < -2x + 4

7. y > -1 8. y + 2x < 4 x > 2

________

9. y > -4x – 3 10. y – 1 < 3x x > 4 y > 3x + 7

11. y – x > 2 12. 2y < -4x – 4 y < 4x + 2 y > -2x – 2


Review QuestionWhat are the possibilities for a solution to a system of equations? A point, No Solution, infinite solutions

What are the possibilities for a solution to a system of inequalities? Region, No Solution, Line

How can the solution be a line? Look at problem #12. If the second inequality was >. DiscussionYesterday, we graphed systems of inequalities by hand. Today, we are going to graph them using the graphing calculators. Why? Easier. Need to know how to use them in the future

SWBAT graph a system of inequalities using a graphing calculator to find the solution set

Example 1: y > 3x + 7 y < -x – 4

Let’s graph it by hand first.

Now, let’s check it with the graphing calculators.Entering the inequality into the graphing calculator:1. Press the “y =” button, then enter the inequality2. Press the arrow key to move the cursor all the way to left, then press enter until you get greater than or less than

Example 2: y > -2x + 3 y + 2x < -8Let’s graph using the graphing calculator.

Empty Set


Example 3: y > 3x + 7 3y + 5x < -8

Let’s graph using the graphing calculator. Make sure to put the 2nd inequality into “y =” form.

Example 4: y > 4 x < -2

Let’s graph it by hand first.

Now, let’s check it with the graphing calculators.What issue do we have? x < -2 doesn’t go into the graphing calculator


Solve each system of inequalities by graphing by hand then confirm your answer on the graphing calculator.

1. y > 3x + 2 2. y > -3x + 7 y < -2x + 4 y < -3x + 2

3. y > 2 4. y + 2x < 3 y > -2

________


Solve each system of inequalities by sketching the solution from the graphing calculator.

5. y > -2x – 1 6. y – 5 < 2x y > 4x + 3 y < 2x + 1

7. y – 3x > 22 8. 5y > 3x – 3 y < 5x + 2 y > -2x – 2

9. y > 3x + 2 10. y > -5 4y < 12x + 8 x < 2

Review QuestionHow do you know what the solution to a system of inequalities is? The intersection of the shaded regions.

SWBAT review for the Unit 6 Test

DiscussionHow do you study for a test? The students either flip through their notebooks at home or do not study at all. So today we are going to study in class.

How should you study for a test? The students should start by listing the topics.

What topics are on the test? List them on the board- Graphing- Substitution- Adding/Subtracting- Multiplication- Graphing Systems of Inequalities

How could you study these topics? Do practice problems; study the topics that you are weak on

Practice Problems Have the students do the following problems. They can do them on the dry erase boards or as an assignment. Have students place dry erase boards on the chalk trough. Have one of the groups explain their solution.

Graph each system of equations. Then determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, name it.

1. y = -x + 4 (-2, 6) 2. 3x + y = 5 Infinite y = 2x + 10 2y – 10 = -6x

3. y + 2x = -1 No Solution y – 4 = -2x

Use substitution, addition/subtraction, or multiplication to solve each system of equations.

4. y = 7 – x (2, 5) 5. x + y = 8 (5, 3) x – y = -3 x – y = 2

6. 2x + 5y = 12 (1, 2) 7. 8x – 6y = 14 (1, -1) x – 6y = -11 6x – 9y = 15

Unit 6 Review

8. 5x – y = 1 (1/4, 1/4) y = -3x + 1


9. y < 3 10. x < 2y y > -x + 2 2x + 3y < 7

11. x > y + 1 2x + y > -4

Write a system of equations. Then solve.12. The difference between the length and width of a rectangle is 7 cm. Find the dimensions of the rectangle if its perimeter is 50 cm. l = 16, w = 9

13. Joey sold 30 peaches from his fruit stand for a total of $7.50. He sold small ones for 20 cents each and large ones for 35 cents each. How many of each kind did he sell? s = 20, l = 10

14. After you do the review problems, pick out one or two topics that you are weak on and find three problems from your notes or homework and do them.


1. Anna burned 15 calories per minute running x minutes and 20 calories per minute hiking for y minutes. She spent a total of 60 minutes running and hiking and burned 1000 calories. The system of equations shown below can be used to determine how much time Anna spent of each exercise.

15x + 20y = 1000 x + y = 60

What is the value of x, the minutes Anna spent running?a. 10 b. 20 c. 30 d. 40

2. Which system is graphed below?

a. 2x + y = -3 b. 4x + y = -3 c. 2x + y = 3 d. 2x + y = 3 y = -2x – 1 -x + y = -3 6x + 3y = 9 y = -2x – 1

3. Solve the system: 3x + 4y = 23 5x + 4y = 25

a. (3, 2) b. (5, 2) c. (1, 5) d. (-10, 6)

Unit 6 Standardized Test Review

4. Several books are on sale at a bookstore. Fiction books cost $4, while non-fiction books cost $6. One day last week 80 books were sold. The total amount of sales was $400. The system of equations shown below can be used to determine how many of each type of book were sold. Let x stand for the number of fiction books and y stand for the number of non-fiction books.

4x + 6y = 400x + y = 80

Which of the following statements is true?

a. There were 30 non-fiction books sold.b. Fiction books cost more than non-fiction books.c. Exactly twice as many fiction books were sold than non-fiction books.d. They sold the same amount of non-fiction and fiction books.

5. The solution set to a system of linear inequalities is graphed below.

Which system of 2 linear inequalities has the solution set shown in the graph?

a. x > 1 b. y > 1 c. y > 1 d. x > 1 y > x + 3 y > -x + 3 y < -x + 3 y > -x + 3

6. The following problems require a detailed explanation of the solution. This should include all calculations and explanations.

The following problem involves systems of equations.

a. What are the three possible solutions to a system of equations? (Explain using sentences and pictures)

b. Make up a system of equations for each one of these possibilities. (Don’t solve them.)

c. Why isn’t it possible to have a system of linear equations that has two solutions?

SWBAT do a cumulative review

DiscussionWhat does cumulative mean?All of the material up to this point.

Our goal is to remember as much mathematics as we can by the end of the year. The best way to do this is to take time and review after each unit. So today we will take time and look back on the first five units.

Does anyone remember what the first six units were about? Let’s figure it out together.1. Pre-Algebra2. Solving Linear Equations3. Functions4. Analyzing Linear Equations 5. Inequalities 6. Systems

Things to Remember:1. Reinforce test taking strategies: guess/check, eliminate possibilities, work backwards, and estimating.2. Reinforce the importance of retaining information from previous units.3. Reinforce connections being made among units.

In-Class Assignment

1. What set of numbers does -5 belong? a. counting b. whole c. integers d. irrationals

2. 4 + 2 = 2 + 4 is an example of what property? a. Commutative b. Associative c. Distributive d. Identity

3. -8.2 + (-4.2) = a. -12.4 b. -3.8 c. 12.4 d. -9.8

4.

a. 20/12 b. 10/12 c. 7/24 d. 2/3

5. (-2.5)(4.7) = a. -9.88 b. -7.2 c. -11.75 d. -5.9

UNIT 6 CUMULATIVE REVIEW

6. 5.18 ÷ 1.4 = a. 4.8 b. 3.2 c. 6.52 d. 3.7

7.

a. – 2/12 b. -1/4 c. -3/4 d. 8/9

8. 33

a. 3 b. 9 c. 12 d. 27 9. = a. 21 b. 29 c. 220.5 d. 87

10. = a. 31.5 b. c. 8 d.

11. 18 – 24 ÷ 12 + 3 a. 15 b. 16 c. 19 d. 20

12. 3x + 4y – 8x + 6y a. 11x +10y b. 5x + 2y c. 5x + 10y d. -5x + 10y

13. 2x + 2 = 14 a. 6 b. -6 c. 8 d. -8

14. 2x + 8 = 5x + 23 a. -5 b. -6 c. No Solution d. Reals

15. 2(x – 3) – 6x = -6 – 4x a. 5 b. 6 c. No Solution d. Reals

16. Solve for y: 4a + 3y = -5x

a. y = 5x – 4a b. c. y = -5x – 4a d. y = -5x – 4a/2

17. Which of the following is a solution to y = 3x + 5 given a domain of {-3, 0, 1} a. (0, 5) b. (1, 2) c. (-3, -1) d. (-3, 7)

18. Which equation is not a linear equation?

a. y = -3x + 2 b. c. y = 5 d. y = x2 + 1

19. Which equation is not a function? a. y = 3x + 7 b. y = 5 c. x = -5 d. y = 1/2x + 2

20. If g(x) = 4x – 3, find g(3). a. 4 b. 5 c. 8 d. 9

21. Write an equation for the following relation: (2, 10) (6, 8) (10, 6)

a. y = -2x b. y = 4x + 12 c. d. y = 2x – 11

22. Write an equation of a line that passes through the points (3, 6) and (4, 8). a. y = x b. y = -2x c. y = 2x + 12 d. y = 2x

23. Write an equation of a line that is perpendicular to and passes thru (-1, 3).

a. y = x b. y = -3x c. y = 3x + 6 d. y = 3x

24. Write an equation of a line that is parallel to y + 2x = -2 and passes thru (3, -2). a. y = -2x + 4 b. y = -2x c. y = -2x + 8 d. y = 2x

25. Write an equation of a line that is perpendicular to x = -3 and passes thru the point (2, -4). a. y = 2 b. y = -4 c. y = 2x d. y = 4

26. Which of the following is a graph of: y = 2x – 5. a. b. c. d.

27. Which of the following is a graph of: y = 3 a. b. c. d.

28. What is the x-intercept of the line y = 4x + 8? a. 4 b. 8 c. -2 d. 2

29. Which of the following is a graph of: y < 2x + 3. a. b. c. d.

30.

a. x < -30 b. x < 30 c. x > 30 d. x > -30

31. |2x + 8| > 14 a. x > 3 or x < -11 b. x > 3 and x < -11 c. x < -11 d. x > 3

32. |4x + 1| > -2 a. x > -3/4 b. x < 1/2 c. No Solution d. Reals

33. Solve the following system of equations. y = x + 2

2x + 3y = 11 a. (0, 2) b. (1, 3) c. (3/2, 1/2) d. (-3, 1)

34. Solve the following system of equations. 3x – y = 10 7x – 2y = 24

a. (0, 5) b. (6, 2) c. (4, 2) d. (-3, 7)

35. Solve the following system of equations. 2x – 6y = 4 2x – 6y = 10

a. No Solution b. Infinite c. (1, 1) d. (-3, 5)

manual62.doc · web viewx + y = 12 3x + 2y = 26 3. x = 4y – 5 (-9, -1) 4. y = 3x + 2 (3, 11) 2x +...

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