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Unit 6: Solving systems

Name: ___________________________________________ Pd: _______

Lesson Book Ch.

TopicHomework Date

due:Grade:

Chapter 5-Quadratic Equations and Functions1 5-5 Zero Product Property and Solving by

Factoringp.266 #1-6 all;p.267 #36,38;46

2 5-8 Solving Quadratics with the Quadratic Formula

p. 289 #2-20 even; find exact solutions (with radicals)

1&2 Quiz: Factoring and Quadratic Formula3 3-1 Solving Systems by Graphing p.119 #1-9 all4 3-2 Solving Systems using Elimination and

Substitutionp. 128 #2-12 even, #18-28 even

3 & 4 Quiz on Solving Linear Systems5 Not

in book

Solving Systems Word Problems p. 157 #6-10, 15; p158 #16-22 even

6 Not in book

Solving nonlinear systems by graphing on the calculator

Packet p. 16-17

Test Review Packet page 18-191 -6 Test

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Lesson 1: The Zero Product Property and Solving By FactoringEquations such as are linear equations, or first degree equations, because the highest power of the variable x is one. A linear, or first degree equation, has one solution because the power of x is one. The graph of a 1st

degree equation is a ____________ .

Equations such as are second degree equations, or quadratic equations. Because the highest power of the variable x is two, these quadratic equations have two solutions. The graph of a 2nd degree equation is a ________________ .

Quadratic equations can be solved by various methods. One of these methods is “by factoring.”

This is all that you need to know: (Zero Product Property)

The key here is to set the equation equal to zero!

Examples- Solve for x: 1. 2. 3.

4. 5. 6.

NOTE: the values you found above for x are the ones which make our y=0;

We call these the __________________, ________________, or __________.

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(Get accustomed to using these words interchangeably).Lesson 1: Classwork:

Solve each of the following quadratic equations by factoring.Hint: Start by getting a zero on the right side!

1. 2.

x-intercepts:_____________ Solutions: __________________

3. 4.

Roots: ________________ Solutions: __________________5. 6.

x-intercepts:_____________ Roots: ________________

7. Do a quick sketch of the graph of #6, but be very specific about your roots.

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Lesson 2: The Quadratic FormulaReview: SOLVE by factoring:1. 2.

Notes: The Quadratic formula

*Always make sure your right side is 0!Solve the following equations using the quadratic formula.

1. 2. 3.

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SOLVE USING THE QUADRATIC FORMULA:

1. 2.

3. 4.

5. 6.

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Warm Up: Solve using Quadratic Formula.1. s2 – 3s – 2 = 0 2. a2 – 3a + 7 = 0 3. 5m2 + 7m = -3

Lesson 3: Solving Systems of Equations by Graphing.

I. System of equations: set of two or more equations that use the same variables.II. The number of solutions depends on the number of times the lines intersect.

Graphs of Equations

Slopes and Intercepts Type Solutions

Lines Intersect Different SlopesAny Intercepts

Consistent, Independent 1

Lines Coincide (same)

Same Slopes and Same Intercepts

Consistent, Dependent infinite

Lines are Parallel Same Slope and Different Intercepts Inconsistent none

III. To solve Systems: 1. Graph each equation (remember: equation needs to be in y = mx + b form)2. See where the graphs INTERSECT…this is the solution3. Use the chart above to describe the type of system.

IV. Examples:1. 2. 3x + y = 8 x + y = 8Put in Slope-Intercept Form:

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

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5

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8

9

–1

–2

–3

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y

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

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8

9

–1

–2

–3

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y

3. 4. Solution = ____________Type of system = _________________

Solution = ____________Type of system = ____________________

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1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

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–6

–7

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–9

y

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

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y

5. 6.

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

–7

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–9

y

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

–7

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–9

y

Solution = ____________Type of system = ____________________

Solution = ____________Type of system = ____________________

Solution = ____________Type of system = ____________________

Solution = ____________Type of system = ____________________

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Graphing Systems – Using Data and graphing on the calculator:

Winning times for the Olympic 400-m run have been decreasing more rapidly for women than for men. Use the data in the table to find linear models for women’s and men’s times. Predict the year in which the women’s winning time could equal that of the men, assuming that current trends continue.

Year 1968 1972 1976 1980 1984 1988 1992 1996 2000

Men’s Time

43.86 44.66 44.26 44.60 44.27 43.87 43.50 43.49 43.84

Women’s Time

52.03 51.08 49.29 48.88 48.83 48.65 48.83 48.25 49.11

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Warm-up - Solve the system of equations by graphing on the calculator. Round your answer to the third decimal place.

1. 2. 3.

Lesson 4: There are three methods for solving systems of linear equations:

(a) ______________________(b) ____________________(c) __________________________

Steps for Substitution:1. Get one variable alone2. Substitute that value for the variable in the other equation3. Solve the equation to get one variable4. Go to your original equation to get the other variable5. CHECK!

Solve the following by substitution:

1. 2. 3.

4. 5.

You Try:

6. 7.

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Steps for Elimination:1. Look at your equations to make sure your variables and constants are in the same order.2. Multiply one or both of the equations so that one variable will cancel out (same number but opposite

sign).3. Get the value of one variable by adding the equations.4. Go back to the original equations to get the other variable.5. CHECK.

Solve the following by elimination:

1. 2. 3.

4. 5.

You Try:

6. 7.

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Warm-up:Solve by Substitution: Solve by Elimination:

1. 2.

Lesson 5: Word Problems involving Systems:1) The sum of two numbers is 64. The bigger number is eight more than three times the smaller. What are the two numbers?

2) Andrew has 22 coins in nickels and dimes. He has $1.55 total. How many nickels and how many dimes does he have?

3) Bill Gates had $370,000 in fifties and hundreds stolen from under his bed. 4300 bills were stolen total. How many of each bill was stolen?

4) The perimeter of a rectangular room is 144 ft. If the room is 12 feet longer than it is wide, find the dimensions of the room.

5) In preparation for baseball season, Coach Hicks has to buy 7 bats and 5 balls for $16.95. Later in the season, he buys 3 bats and 6 balls for $13.05. What is the price of a baseball? What is the price of a baseball bat?

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6) The sum of two numbers is five times their difference. If one exceeds the other by seven, what are the numbers?

7) The width of a rectangle is 10 feet less than twice the length. The perimeter is 112 feet. Find the width.

8) Sue’s mother is seven years less than three times Sue’s age. Ten years ago, her mother was 3 years more than five times Sue’s age. How old are Sue and her mother currently?

9) Slimy Joe’s Ice Cream Parlor sells half as many ice cream cones as milkshakes. The cost of an ice cream cone is $1.35 and the cost of a milkshake is $3. If the shop took in $330.75 on Sunday, how many of each was sold?

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Warm-up: To pay your monthly bills, you can either open a checking account or use an online banking service. A local bank charges $3 per month and $0.40 per check, while an online service charges a flat fee of $9 per month.a) Write and graph a system of linear equations to model the cost, y, of each service for x bills that you need to pay monthly.b) Find the point of intersection of the two lines. What does this answer represent?c) If you pay about 12 bills per month, which service should you choose? Explain.

Lesson 6: Solving and Graphing Systems of Linear and Nonlinear Equations with the Graphing Calculator

1. 2.

-9 9

-9

9

x

y

-9 9

-9

9

x

y

SOLUTION: _______________________ SOLUTION: _______________________

Exercise #3: Solve each equation for y, then graph on the calculator to determine the solutions.

(a) (b)

-9 9

-9

9

x

y

-9 9

-9

9

x

y

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SOLUTION: _______________________ SOLUTION: _______________________

(c) (d)

-9 9

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-9 9

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x

y

SOLUTION: _______________________ SOLUTION: _______________________

4. 5.

-9 9

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x

y

-9 9

-9

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x

y

SOLUTION: _______________________ SOLUTION: _______________________

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Exercise #6: Solve each equation for y, then graph on the calculator to determine the solutions. Round to the nearest hundredth when necessary.

(a) (b)

-9 9

-9

9

x

y

-9 9

-9

9

x

y

SOLUTION: _______________________ SOLUTION: _______________________

(c) (d)

-9 9

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x

y

-9 9

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9

x

y

SOLUTION: _______________________ SOLUTION: _______________________

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HOMEWORK:Solving and Graphing Systems of Linear and Nonlinear Equations with the Graphing Calculator

1. 2.

-9 9

-9

9

x

y

-9 9

-9

9

x

y

SOLUTION: _______________________ SOLUTION: _______________________

3. 4.

-9 9

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x

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x

y

SOLUTION: _______________________ SOLUTION: _______________________

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5. 6.

-9 9

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x

y

SOLUTION: _______________________ SOLUTION: _______________________

7. 8.

-9 9

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x

y

SOLUTION: _______________________ SOLUTION: _______________________

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Test Review: Solve the following by factoring.

1. 2. 3.

4. 5. 6.

Solve the following quadratic equations (how many answers?) by factoring. Show work!

7. 8. 9.

10. 11. 12.

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Solving and Graphing Systems of Linear and Nonlinear Equations with the Graphing Calculator

1. 2.

-9 9

-9

9

x

y

-9 9

-9

9

x

y

SOLUTION: _______________________ SOLUTION: _______________________