Algebra 1 - Welcome to Mrs. Smith's Math Class



Unit 1 – Linear Functions

|Day |Date |Lesson |Write your homework here! |

|Mon |8/29 | | |

| | |Classroom Policy; Pre-Test; Linear Equations Review | |

|Tues |8/30 |Review of Linear Functions | |

|Wed |8/31 | | |

| | |Linear Review continued (Writing Equations) | |

|Thurs |9/1 |More with Linear Functions | |

| | |Solving Systems of Equations by Graphing | |

|Fri |9/2 |Quiz 1 (Linear Review) | |

| | |Solving Systems by Substitution & Elimination | |

|Mon |9/5 |Holiday – No School | |

|Tues |9/6 |Systems continued | |

|Wed |9/7 |Review for TEST | |

|Thurs |9/8 |TEST – Unit 1 | |

| | |Graphing Linear Inequalities/Systems of Inequalities | |

|Fri |9/9 |Early Release | |

| | |More with Inequalities; Intro to Linear Programming | |

|Mon |9/12 |Linear Programming | |

|Tues |9/13 |Linear Programming | |

|Wed |9/14 |Quiz 2 (Linear Inequalities and Linear Programming) | |

Math 3 Name ________________________________

Review of Solving Linear Equations

The basic concept to solving a linear equation is to isolate the variable by getting rid of parentheses and/or fractions and using various inverse operations.

There are 3 types of solutions for linear equations:

• One solution (or x=?) – there is variable = constant

• Infinitely many solutions – where the variable cancels and you are left with a TRUE statement (Ex: 2 = 2)

• No solution – where the variable cancels out and you are left with a FALSE statement (Ex: 1= 4)

Solve each equation.

[pic]

Math 3 Name __________________________________

Unit 1, Lesson 1– Review of Linear Functions Date ___________________________

Linear Function: a function whose graph is a line

Linear functions do NOT contain:

• Variables with exponents

• Variables in the denominator of a fraction

• The product of variables

Slope-intercept form: [pic] where m = __________________ and b = __________________

Standard form: [pic]

Graph each equation:

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

Slope of a non-vertical line = ______________________ = __________

Formula for Slope: m =

Find the slope:

6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. 11. [pic]

12. [pic] 13. [pic]

**The slope of a horizontal line is always equal to zero. Remember “hoy”!

**The slope of a vertical line is always undefined. Remember “vernox”

Parallel vs Perpendicular Lines

Parallel Lines have ____________________ slopes.

Examples:

Perpendicular Lines have slopes that are _______________________________.

Examples:

Linear Review Practice #1 (2 Pages)

I. Graphs - Graph each of the following on the given coordinate plane

[pic]

II. Slope

Math 3 Name __________________________________

Unit 1, Lesson 2– Review of Linear Functions continued Date ___________________________

Forms of Linear Equations

Slope-intercept Form: [pic]

Point-Slope Form: [pic]

Standard Form: [pic]

Write the equation of the line given the following information:

1. Slope & y-intercept (m & b)

a. m = -2, b = 8 b. m = [pic] , b = -4

2. Slope & a point – Use point-slope formula

a. m = 3, (-1, 5) b. m = 2, (4, -2) c. m = [pic], (5, 6)

3. Two points – Find slope & then pick one of the points and use point-slope formula

a. (-2, -2), (4, 2) b. (5, 0), (-3, 2) c. (5, 1), (4, -3)

Parallel & Perpendicular Slope

Parallel lines have _______________ slopes.

Perpendicular lines have slopes that are __________________________.

Write the equation of the line using the given information.

4. Passes through the point (2, 1) and is parallel to the line [pic]

5. Passes through the point (-1, 3) and is perpendicular to the line [pic]

6. Passes through the point (-3, -1) and is perpendicular to the line [pic]

7. Passes through the point (-2, 5) and is vertical 8. Passes through the point (-2, 5) and is horizontal

Linear Review Practice #2 (2 Pages)

USE A SEPARATE SHEET OF PAPER!

Write the equation of the line in slope-intercept form given the slope and the y-intercept.

1. [pic] 2. [pic] 3. [pic]

Write the equation of the line in slope-intercept form given the slope and passes through the given point.

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

Write equation of the line in slope-intercept form that passes through the two points.

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

Write an equation in slope-intercept form that satisfies each condition.

16. passes through [pic] and is parallel to the graph of [pic]

17. passes through [pic] and is perpendicular to the graph of [pic]

18. passes through [pic] and is parallel to the graph of [pic]

19. passes through [pic] and is perpendicular to the graph of [pic]

20. passes through [pic] and is perpendicular to the graph of [pic]

21. passes through [pic] and is parallel to the graph of [pic]

22. passes through [pic] and is parallel to the line that passes through [pic]and [pic]

Name ________________________

Mix, Freeze, Pair

Linear Equations

1. Point 1_________________ Point 2____________________

Slope: Graph:

Equation:

2. Point 1_________________ Point 2____________________

Slope: Graph:

Equation:

3. Point 1_________________ Point 2____________________

Slope: Graph:

Equation:

[pic]

Check Understanding - Linear Review

Find the slope, then graph .

1. (5x ( 2y = (6 Slope = _______

2. The slope of a vertical line is ____________________________.

3. The slope of a horizontal line is __________________________.

Write the equation of the line in slope-intercept form using the given information.

4. slope =[pic]; (10, 4) 5. ((2, 3) and (2, 9) 6. Passes through (3,2) and perpendicular to [pic]

Check Understanding - Linear Review

Find the slope, then graph .

1. (5x ( 2y = (6 Slope = _______

2. The slope of a vertical line is ____________________________.

3. The slope of a horizontal line is __________________________.

Write the equation of the line in slope-intercept form using the given information.

4. slope =[pic]; (10, 4) 5. ((2, 3) and (2, 9) 6. Passes through (3,2) and perpendicular to [pic]

Math 3 Name __________________________________

Unit 1, Lesson 3– Systems of Equations Date ___________________________

A system of equations is a set of 2 or more equations that use the same variables. To solve a system of equations we find the point(s) of intersection.

There are 3 methods: 1) Graphing 2) Substitution 3) Elimination

I. Solving Systems by Graphing

To solve by graphing:

Solve the following systems by graphing (using your calculator if possible):

1. [pic] 2. [pic]

3. [pic] 4. [pic]

II. Solving Systems by Substitution

Solve the following systems by substitution:

5. [pic] 6. [pic]

7. [pic] 8. [pic]

III. Solving Systems by Elimination

Another method for solving systems is called the elimination method or the addition or subtraction method.  In this way of solving systems of equations, one variable is eliminated by adding or subtracting the equations.

Example 3: When adding two equations, you make sure the parts are lined up and then you basically add all parts of them.  Say you had the equations:

4x + 5y = 14

-4x - 3y = -10

Adding them would give 2y = 4

4x + 5y = 14

-4x - 3y = -10

         2y = 4

As you can see, the 4x and -4x cancelled out, therefore eliminating the variable x, leaving an equation with only one variable (y), able to be solved.

(continued on next page)

2y = 4

y = 2

Now that you have a value for y, you must find one for x.  To do this, just substitute the value for y into either original equation, and solve it for x

4x + 5(2) = 14

4x + 10 = 14

4x = 4

x = 1

Your solution for these two equations is (1, 2).

Notice that only because 4x and -4x, when added, produce 0 (cancel out), the equation can be solved.  Their coefficients are opposites of each other.  That is why it worked.

But suppose the coefficients on the variables will not eliminate when added:

Example 4: Try this one -

6x - 2y = 18

6x - 7y = 3

These two equations, if added, do not help.  Instead, you can subtract them.  Subtraction is just addition of the opposite, so change EVERY sign in one equation, and add it to the other.

Work for example 2:

But what if you want to solve a system using this method where no coefficients are the same or opposites of each other? We must pick a variable to eliminate and multiply one or both of the equations to make the coefficients the same number with opposite signs.

Example 5:

3x + 5y = 12

4x - 3y = -13

More practice. Solve the following systems using the elimination method:

9. [pic] 10. [pic]

11. [pic] 12. [pic]

Warm-up – Lessons 1 – 3 Name __________________________________

1. Graph the following:

a. [pic] b. [pic]

c. [pic] d. [pic]

2. Are the lines [pic] and [pic] parallel, perpendicular or neither?

Write the equation of the line in slope-intercept form using the given information.

3. slope =[pic]; (5, 2) 4. ((3, 5) and (1, 4)

5. Solve the following system by graphing (using your calculator): [pic]

Kuta worksheet on solving systems all 3 methods

[pic]

More Practice – Application with Systems of Equations

SOLVE ON A SEPARATE SHEET!

Steps to Solving Linear Systems as Real-Life Models

1. The sum of two numbers is 36 and their difference is 8. Find each of the numbers.

2. A travel agent offers two package vacation plans. The first plan costs $360 and includes 3 days at a hotel and a rental car for 2 days. The second plan costs $500 and includes 4 days at a hotel and a rental car for 3 days. The daily charge for the room is the same under each plan, as is the daily charge for the car. Find the cost per day for the room and for the car.

3. The perimeter of a rectangle is 32 cm. The length is 1 cm more than twice the width. Find the dimensions of the rectangle.

4. Rick has 11 coins in his pocket worth $2.30. Some are dimes and some are quarters. How many of each coin does he have?

5. For 1980 through 1990, the coal production in the world, C (in millions of metric tons), and crude petroleum production, P (in millions of metric tons), can be modeled by the following equations. [pic], where [pic] represents 1980. During which year were the production levels equal?

6. Mrs. Lopez bought 3 containers of yogurt and 5 grapefruits for $2.79. The following week she bought 4 containers of yogurt and 7 grapefruits for $3.81, paying the same prices. Find the cost of a container of yogurt and the cost of each grapefruit.

7. A company ordered two types of parts, brass and steel. A shipment containing 3 brass and 10 steel pars costs $48. A second shipment containing 7 brass and 4 steel parts costs $54. Find the cost of a brass part and the cost of a steel part.

8. The sum of two numbers is 108 and their difference is 16. Find each of the numbers.

9. The perimeter of a rectangle is 56 cm. The length of the rectangle is 2 cm more than the width. Find the dimensions of the rectangle.

10. The sum of 4 times Joan’s age and 3 times Jim’s age is 47. Jim is 1 year less than twice as old as Joan. Find each of their ages.

11. The cost of 8 adult’s tickets and 7 children’s tickets is $82.45. The cost of 6 adult’s tickets and 9 children’s tickets is $78.15. Find the cost of each adult’s ticket and each child’s ticket.

12. The cost of 3 boxes of envelopes and 4 boxes of notepaper is $13.25. Two boxes of envelopes and 6 boxes of note paper cost $17. Find the cost of each box of envelopes and each box of notepaper.

13. A group of 42 people go to an amusement park. The admission fee for adults is $16. The admission fee for children is $12. The group spent $568 to get into the park. How many adults and how many children were in the group?

14. The drama club at Lincoln High School sells hot chocolate and coffee at the school’s football games to make money for a special trip. At one game, they sold $200 worth of hot drinks. They need to report how many of each type of drink they sold for their club records. Macha knows that they used 295 cups that night. If hot chocolate sells for 75¢and coffee sells for 50¢, how many of each type of hot drink did they sell?

15. Lewis Carroll was a mathematician and an author. He used mathematical logic when writing his stories. In his book, Through the Looking Glass, there is a conversation between Tweedledee and Tweedledum. Tweedledum says, “The sum of your weight and twice mine is 361 pounds.” Tweedledee answers, “Contrariwise, the sum of your weigh and twice mine is 362 pounds.” What are the weights of Tweedledee and Tweedledum?

16. At a local amusement park it costs as fixed rate to park the car and a certain fee per person. One car entered with 3 passengers and paid $22. The next car had 5 passengers and paid $34. Find the cost of parking and the cost per person.

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

18. The length of Sally’s garden is 4 meters greater than 3 times the width. The perimeter of her garden is 72 meters. What are the dimensions of Sally’s garden?

19. A thirty-minute television show consists of commercials and the program. If the time spent on commercials is fourteen minutes less than the time spent on the program. How many minutes of commercials are there?

20. The measures of the two acute angles of a right triangle differ by 18 degrees. What are their measures?

21. Hal makes tables and chairs. Each chair requires 4 feet of oak and 3 feet of pine, while each table requires 8 feet of oak and 2 feet of pine. Hal has 52 feet of oak and 23 feet of pine. How many chairs and tables can he build?

Math 3 Name_________________________________

Unit 1 Review Date ________________________ Per _____

Find the slope of the line passing through each pair of points for questions 1 & 2.

1. [pic] and [pic] ______________

2. [pic] and [pic] ______________

3. Find the slope of a line that is parallel to[pic]. ______________

4. Find the slope of a line that is perpendicular to [pic] ______________

Write the equation of the line in slope-intercept form with the given conditions.

5. passes through[pic]; m = 4 _____________________

6. passes through [pic] and [pic] _____________________

7. passes through (-3, 4) and parallel to [pic] _____________________

8. passes through (1, 4) and perpendicular to [pic] _____________________

Graph each function on the provided graph. Label your axes! Be specific with your points!

9. [pic] 10. [pic]

Solve ALL problems on a SEPARATE SHEET!

11. Solve the given systems by graphing. Use your calculator.

a) b)

22.

12. Solve the given systems by substitution.

a) b)

13. Solve the given systems by elimination.

a) b)

Set up a system of equations and solve the following problems. You must show system for credit.

14. Suppose you have 28 coins that are all dimes and quarters. The total value of the coins is $5.35. Write a system of equations that models this situation and solve the system to find the number of quarters and the number of dimes.

15. You have $22 in your bank account and deposit $11.50 each week. At the same time, your cousin has $218 and is withdrawing $13 each week. When will your accounts have the same balance?

16. The senior classes at Siloam High School and Gentry High School planned to separate trips to New York City. The senior class Gentry rented and filled 1 van and 6 buses with 372 students. Siloam rented and filled 4 vans and 12 buses with 780 students. Each van and each bus carried the same number of students. How many students can a van carry? How many students can a bus carry?

17. The owner of a movie theater was counting the money from 1 day's ticket sales. He knew that a total of 150 tickets were sold. Adult tickets cost $7.50 each and the children's tickets cost $4.75 each. If the total receipts for the day were $891.25, how many of each kind of ticket were sold?

18. The perimeter of a rectangle is 48 cm. The length of the rectangle is 6 cm more than the width. Find the dimensions of the rectangle.

-----------------------

Assign the variables.

Write an algebraic model.

Answer the question.

No guessing!

Solve the model.

[pic]

2x + 4y = 12

x + y = 2

x + 3y = 7

2x – 4y = 24

[pic]

2x – 3y = -1

3x + 4y = 8

[pic]

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