Linear Systems. Name: Unit 4 Beaumont Middle School 8th Grade, Introduction to Algebra

Transcription

1 Unit 4 Beaumont Middle School 8th Grade, Introduction to Algebra Name: Linear Sstems I can graph using slope and -intercept in form. I can graph using and intercepts in form. I can write equations for word problems in I can graph sstems in form and analzing intersection. I can write equations for word problems in form. I can graph sstems in form and analzing intersection. I can solve sstems b substitution. I can solve sstems b elimination.

2 Graphing Lines, ~~ Unit 4, Page 2 ~~ using -intercept and slope I can graph lines in form using the slope and -intercept. The formula = m + b is said to be a linear function. That is the graph of this function will be a straight line on the (, ) plane. One could epress this as a formal function definition with notation such as: f() = m + b Since we will be graphing (, ) points, though, we will do our thinking with the ' = m + b' form for a while. When the function for a line is epressed this wa, we call it the 'slope-intercept form'. Where is the slope? The slope of the line is the variable m. The slope describes the slant of the line. Where is the intercept? B 'intercept' we mean '-intercept'. The -intercept is held b the variable b. The -intercept is the point where the line crosses the -ais. If ou know the slope for the line... and ou know the - intercept for the line... then ou can write the slope-intercept equation for the line.

3 ~~ Unit 4, Page 3 ~~ Graph the following lines using the -intercept and slope. 1) 2) 3) -intercept: slope: -intercept: slope: b: m: 4) 5) + 4 6) -intercept: slope b: m: -intercept: slope 7) 8) 9) b: m: -intercept: slope b: m: Homework is continued

4 ~~ Unit 4, Page 4 ~~ 1) 11) 12 ) -intercept: slope b: m: -intercept: slope 13) 14) 15) -intercept: slope b: m: -intercept: slope 16) 17) 18) b: m: -intercept: slope b: m:

5 Graphing L ines, ~~ Unit 4, Page 5 ~~ with and intercepts I can graph lines in using the and -intercepts. form Equations that are written in form are easier to graph using the -intercept and -intercepts. Before we begin, let's see what standard form looks like. What is Standard Form? Now let's review what the term intercepts means. An intercept is where our line crosses an ais. We have an intercept and a intercept. The point where. the line touches the ais is called the intercept.. The point where the line touches the ais is called the intercept. If we can find the points where the line crosses the and ais, then we would have two points and we'd be able to draw a line. When equations are written in standard form, it is prett eas to find the intercepts. Take a look at this diagram, as it will help ou to understand the process. Now, let's appl this. Just remember: To find the intercept: Let = To find the intercept: Let =

6 ~~ Unit 4, Page 6 ~~ Eample 1 Use the and intercepts to graph the equations. 1) 2) 3) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, )

7 ~~ Unit 4, Page 7 ~~ On Your Own Use the and intercepts to graph the equations. 1) 2) 3) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) 4) 5) 6) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) Homework is continued

8 ~~ Unit 4, Page 8 ~~ 7) 8) 9) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) 1) 11) 12) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) 13) 14) 15) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, )

9 Equations of Lines (slope-intercept form) ~~ Unit 4, Page 9 ~~ I can write and evaluate an equation in slope-intercept form given a real life situation. When ou have a real world (word problem) that requires ou to write an equation in slope intercept form, there are two things that ou want to look for: 1. A Rate. The rate is our slope in the problem. The following are eamples of a rate $3 per da $2 an hour 6 mph This number is alwas related to the -value. Per is a ke word that is often associated with slope. 2. A Flat Fee. A flat fee or starting value is our -intercept. This value is a constant. It never changes. 2 m/s $6 a minute 45 words per minute Use the chart below to help ou organize our information as ou analze each word problem. This will help ou to write our equation! Flat Fee (starting #) b (-intercept)? Rate m (slope)? Take a look at the eamples below to better clarif how this chart can help ou! Eample 1 You are visiting Baltimore MD, and a tai compan charges a flat fee of $3. for using the tai and an additional $.75 per mile. Write an equation that ou could use to find the cost of a tai ride in Baltimore, MD. Let represent the number of miles and represent the total cost. How much would a tai ride for 8 miles cost? Flat Fee (starting #) Rate b (-intercept) m (slope) = m + b = ( ) + The equation could be used to find the cost of a tai ride in Baltimore, MD is To find out the cost for an 8 mile ride, substitute 8 for. = ( ) + A tai ride would cost for 8 miles.

10 ~~ Unit 4, Page 1 ~~ Eample 2 A plumber charges a fee of $12 to make a house call. He also charges $1. an hour for labor. Write an equation that ou could use to find the amount a plumber charges for a house call based on the number of hours of labor. Let represent the number of hours of labor and represent the total cost. How much would a house call cost that requires 2.5 hours of labor? Flat Fee (starting #) Rate b (-intercept) m (slope) = m + b = ( ) + The equation could be used to find the amount a plumber charges is To find out the cost for the 2.5 hours, substitute 2.5 for. = ( ) + A plumber would cost for 2.5 hours. Your Turn 1. Hannah's electricit compan charges her $.1 per kwh (kilowatt-hour) of electricit, plus a basic connection charge of $15. each month. Write a linear function that models her monthl electricit bill as a function of electricit usage. Let represent the cost and represent the amount of electricit. How much would her bill be if she used 5kWh of electricit? Flat Fee (starting #) Rate b (-intercept) m (slope) = m + b = ( ) + The equation could be used to find the charge on her electric bill is To find out the cost for the electricit, substitute 5 for. = ( ) + A bill would be for 5kWh. Homework is continued

11 ~~ Unit 4, Page 11 ~~ 2. Joe is throwing a part. The clubhouse charges $5 to rent the space and $25 per person. Write a linear function that models the total bill as a function of the number of guests. Let represent the cost and represent the number of people attending. How much would the bill be if there were 4 attendees? Flat Fee (starting #) b (-intercept) Rate m (slope) = m + b = ( ) + The equation could be used to find the charge on Joe s part bill is To find out the cost for the clubhouse, substitute 4 for. = ( ) + A bill would be for 4 people 3. Savannah is driving on a trip. She is going an average speed of 7mph. She has alread gone 1 miles toda. Write a linear function that models the total distance as a function of the number of hours left to travel. Let represent the distance and represent the number of hours. How man miles would she have travelled in 6 more hours. Flat Fee (starting #) b (-intercept) Rate m (slope) = m + b = ( ) + The equation could be used to find the distance travelled is To find out the distance travelled in 6 more hours, substitute 6 for. = ( ) + The distance would be for 6 hours. 4. Jordan is buing a new TV. She can make a down pament of $1, and then will pa $6 per month. Write a linear function that models the total amount paid as a function of the number of months. Let represent the amount paid and represent the number of months. How much mone will Jordan have paid in 12 months? Flat Fee (starting #) Rate b (-intercept) m (slope) = m + b = ( ) + The equation could be used to find the total paid is To find out the total paid in 12 months, substitute 12 for. = ( ) + The total paid would be for 12 months. Homework is continued

12 ~~ Unit 4, Page 12 ~~ 5. Kallie is conditioning for tr-outs. She has alread run 1 miles. She will run 2 miles per da. Write a linear function that models the total she has run as a function of the number of das. Let represent the total number of miles and represent the number of das. How man miles will Kallie have run in 2 das? Flat Fee (starting #) Rate b (-intercept) m (slope) = m + b = ( ) + The equation could be used to find the total number of miles is To find out the total number of miles, substitute 2 for. = ( ) + The total run would be in 2 das. 6. Bethan is renting a cabin in Tennessee. The charge a $2 cleaning fee and $1 per night. Write a linear function that models the total amount charged as a function of the number of nights of the vacation. Let represent the total charged and represent the number of nights. How much would be charged for a 4 night sta? Flat Fee (starting #) Rate b (-intercept) m (slope) = m + b = ( ) + The equation could be used to find the total amount charged is To find out the total amount charged, substitute 4 for. = ( ) + The total bill would be for 4 nights. Homework is continued

13 ~~ Unit 4, Page 13 ~~ Review. Simplif the following epressions. Show work without a calculator. 1) 2) 3) 4 ) Evaluate the following epressions if = 4, = 2, and z = 1. Show work. 5) 6) 7) 8) Solve each of the following. Show work. 9) 1) 11) 12) 13) 14) 15) 16) 17)

14 ~~ Unit 4, Page 14 ~~ Graphs of Linear Sstems (slope-intercept form; ) I can create a math model for a real life situation using sstem of equations in slope intercept form and a graph. Fail Safe Super Locks

15 ~~ Unit 4, Page 15 ~~ A. Use the graphs to estimate the answers to these questions. 1. For what number of das will the costs for the two companies be the same? What is the cost? 2. For what number of das will Super Locks cost less than Fail Safe? 3. For what number of das will Superlocks cost more than Fail Safe? 4. For what number of das will Super Locks cost less than $6? 5. What is the cost of one ear of service from Fail Safe? B. For each compan, write an equation for the cost, c, for d das of securit services. Super Locks: Fail Safe: Sometimes it is easier to graph equations of lines using two points. The following problem asks ou to fill in the first and last values for and find the -values. Sam needs to rent a car for a one-week trip in Oregon. He is considering two companies. A+ Auto Rental charges $16 plus $.1 per mile. Zipp Auto Rental charges $8 plus $.2 per mile. Define our variables: rental cost: Miles driven: Equation for A+ Auto Rental: Equation for Zipp Auto Rental: b. Complete the missing values in the table and then graph the equations. (include titles) A+ Auto Rental Miles 1 Cost Zipp Auto Rental Miles 1 Cost 1) Approimate the point of intersection: 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if more miles are travelled and what it means if fewer miles are traveled.)

16 Eample 1: Tai Compan A You are visiting Baltimore MD, and Tai Compan A charges a flat fee of $3. for using the tai and an additional $.75 per mile. Write an equation that ou could use to find the cost of a tai ride in Baltimore, MD. = the # of miles = the cost. Equation: ~~ Unit 4, Page 16 ~~ Graph both equations on the following grid. Use an interval of 1 on both aes. Total Cost Eample 2: Tai A vs. Tai B Number of Miles Brad the Plumber Brad, a plumber, charges a fee of $12 to make a house call. He also charges $1. an hour for labor. Write an equation that ou could use to find the amount Brad charges for a house call based on the number of hours of labor. = # of hours = the cost. Equation: Tai Compan B You are visiting Baltimore MD, and Tai Compan B charges a flat fee of $5 for using the tai and an additional $.5 per mile. Write an equation that ou could use to find the cost of a tai ride in Baltimore, MD. = the # of miles and = the cost. Equation: 1) Name the point of intersection: 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if more miles are travelled and what it means if less miles are traveled.) Valeria the Plumber Valeria, a plumber, charges a fee of $1 to make a house call. She also charges $15. an hour for labor. Write a equation that ou could use to find the amount Valeria charges for a house call based on the number of hours of labor. = # of hours = the cost. Equation: Graph both equations on the following grid. Use an interval of 1 on the -ais and 2 on the -ais. Plumber Bills Total Cost Number of Hours 8 4 1) Name the point of intersection: 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if more hours are needed and what it means if fewer hours are needed.) 1 5

17 ~~ Unit 4, Page 17 ~~ On our own; #1 Hannah s Electricit Hannah's electricit compan charges her $.1 per kwh (kilowatt-hour) of electricit, plus a basic connection charge of $15. per month. Write a linear function that models her monthl electricit bill as a function of electricit usage. Kerr s Electricit Kerr's electricit compan charges her $.15 per kwh (kilowatt-hour) of electricit, plus a basic connection charge of $1. per month. Write a linear function that models her monthl electricit bill as a function of electricit usage. = the cost and = kwh of electricit. Equation: 2 = the cost and = kwh of electricit. Equation: 2 Graph both equations on the following grid. Use an interval of 2 on the -ais and 5 on the -ais. Electricit 1) Name the point of intersection: Total Cost 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if more kwh are needed and what it means if fewer kwh are needed.) Number of kwh On our own; #2 Joe s Part Joe is throwing a part. The clubhouse charges $5 to rent the space and $25 per person. = cost and = # of people Equation: Jack s Part Jack is throwing a part. The clubhouse charges $6 to rent the space and $15 per person. = cost and = # of people Equation: Graph both equations on the following grid. Use an interval of 2 on the -ais and 1 on the -ais. Part Time 2 1) Name the point of intersection: 2 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if more people attend and what it means if fewer people attend.) Total Cost Number of People Homework is continued

18 ~~ Unit 4, Page 18 ~~ On our own; #3 Savannah s Trip Savannah is driving on a trip. She is going an average speed of 7mph. She has alread gone 1 miles toda. Am s Trip Am is driving on a trip. She is going an average speed of 5mph. She has alread gone 2 miles toda. = distance and = # hours = distance and = # hours Equation: 1 Equation: 1 Total Distance Graph both equations on the following grid. Use an interval of 1 on the -ais and 1 on the -ais. Travelling with Savannah and Am 1) Name the point of intersection: 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if more hours are travelled and what it means if fewer hours are travelled.) Number of Hours On our own; #4 Jordan s TV Jordan is buing a new TV. She can make a down pament of $1, and then will pa $6 per month. = $ paid = # of months Equation: 1 Perla s TV Perla is buing a new TV. She can make a down pament of $2, and then will pa $4 per month. = $ paid = # of months Equation: 1 Graph both equations on the following grid. Use an interval of 1 on the -ais and 1 on the -ais. Paing for a TV 1) Name the point of intersection: 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if the pa more months and what it means if the pa fewer months.) Total Paid Number of Months

19 ~~ Unit 4, Page 19 ~~ Equations of Lines (standard form, ) I can write a sstem of equations in standard form given a real life situation. We've studied word problems that allow for ou to write an equation in slope intercept form. How do we know when a problem should be solved using an equation written in standard form? In standard form, there appears to be 2 rates! These two numbers are the number per and the number per. Each of these is multiplied to and, respectivel. There is no beginning amount, nor are there points given. However, there ma be a TOTAL involved. In this case, the equation can be written in A B C form with C being the total amount. Neither variable is dependent on the other in this case! As ou are reading and analzing the word problem, if ou find that ou can set up two addition problems, and ou have two set totals (constant) one tells ou the value and the other the total number, then ou will be able to write equations in standard form. Eample 1: You are running a concession stand at the basketball game. You sell hotdogs for $1 and sodas for $2. Let our variables be the number of each of the items. : # of hotdogs : # of sodas You sold a total of 12 items. At the end of the night, ou made $2. Write an equation for the number of items ou sold: Write an equation for the value of the items ou sold: Eample 2: Beaumont is sponsoring a pancake dinner to raise mone for a field trip. Each adult ticket will cost $2 and each child s ticket will cost $1. Let our variables be the number of each tpe of ticket. : # of adults : # of children You estimate a total of 7 tickets to be sold. At the end of the night, ou made $9. Write an equation for the number of tickets ou sold: Write an equation for the value of the tickets ou sold: Your turn. 1) A test has multiple choice questions worth 2 points apiece and short answer questions worth 4 points apiece. Let our variables be the # of each tpe of question. : # of multiple choice; : # of short answer. There are a total of 3 questions. The test is worth a total of 1 points. Write an equation for the number of questions that ma be on the test: Write an equation for the value of the test questions: Homework is continued

20 2) Justin has saved five dollar bills and singles. ~~ Unit 4, Page 2 ~~ Let our variables be the # of each tpe of bills. : # of $5 bills; : # of $1. Justin has a total of 35 bills. His savings are worth a total of $75. Write an equation for the number of bills Justice has. Write an equation for the value of the bills: 3) Claire bought sandwiches and drinks at the ballgame. The sandwiches cost $4 each and the drinks were $2 each. Let our variables be the number of each tpe of item. : # of sandwiches; : # of drinks Claire bought 9 items for a total of $28. Write an equation for the number of items Claire bought: Write an equation for the value of the items: 4) The store at which Michael usuall shops is having a sale. Roast beef costs $4 a pound and shrimp costs $1 a pound. He bought 16 pounds of meat for a total cost of $1. Let our variables be the #r of pounds of each tpe of meat: : # of Lbs of roast beef; : # Lbs of shrimp Write an equation for the number of pounds that Michael bought: Write an equation for the value of the meat: 5) It will take 2 points to make the plaoffs, the hocke team coach told the plaers. We get 2 points for a win and 1 point for a tie. The team has 12 games left in the season. Let our variables be the # of each tpe of outcome: : # of wins; : # of ties Write a sstem of equations: Homework is continued

21 ~~ Unit 4, Page 21 ~~ 6) You are in charge of buing the hamburger and chicken for a part. You have $6 to spend. The hamburger costs $2 per pound and chicken is $3 per pound. You bought 25 pounds of meat. Define our variables: and Write a sstem of equations: 7) You are buing $48 worth of lawn seed that consists of two tpes of seed. One tpe is a quickgrowing re grass that costs $4 per pound, and the other tpe is a higher-qualit seed that costs $6 per pound. You need a total of 11 pounds of seed. Define our variables: and Write a sstem of equations: 8) Your grandmother made 24 oz. of jell. You have two tpes of jars. The smaller holds 1 oz. And the larger holds 12 oz. Your grandmother wants to fi 22 jars. Define our variables: and Write a sstem of equations: 9) You are buing $3 worth of birdseed that consists of two tpes of seed. Thistle seed attracts finches and costs $2 per pound. Dark oil sunflower seed attracts man kinds of sunbirds and costs $1.5 per pound You are buing 18 pounds of birdseed. Define our variables: and Write a sstem of equations:

22 # student members ~~ Unit 4, Page 22 ~~ Graphs of Linear Sstems (Standard Form: ) I can create a math model for a real life situation using sstem of equations in standard form and a graph. 1. At a school band concert, Christopher and Celine sell memberships for the band s booster club. An adult membership costs $1, and a student membership costs $5. At the end of the evening, the students had sold 5 memberships for a total of $4. The club president wants to know how man of the new members are adults and how man are students. A. Let stand for the number of $1 adult memberships and for the number of $5 student memberships. 1. What equation relates and to the $4 income? 2. Give two solutions for our equation from part (1). and 3. What equation relates and to the total of 5 new members? Are the solutions ou found in part (2) also solutions of this equation? B. 1. Graph the two equations from Question A on the grid. These charts will help ou find the and intercepts. Income Equation: # of Adults # of Students # of Members Equation: # of Adults # of Students 2. Estimate the coordinates of the point where the graphs intersect. Eplain what the coordinates tell ou about the situation. (Include both values and what it means to both equations.) In Question A, ou wrote a sstem of equations. One equation represents all (, ) pairs that give ou a total income of $4, and the other represents all (, ) pairs that give ou a total of 5 memberships. The coordinates of the intersection point satisf both equations, or conditions. These coordinates are the solution to the sstem.

23 ~~ Unit 4, Page 23 ~~ 2. For a fundraiser, students sell calendars and posters. Each calendar will profit them $3 and each poster will profit them $2. Their goal is to earn $6. 25 items have been donated b a generous corporation. p = #of posters a. Write an equation to represent earning the $6. b. Write an equation to represent the donation of 25 items. c. Graph both equations on the same coordinate grid. (c, p) Use an interval of 25 on the -ais and 5 on the -ais. These charts will help ou find the and intercepts. c = # of calendars Value Equation: Posters # of Items Equation: 5 d. State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. (Include both values and what it means to both conditions.) 3. Neema has a collection of quarters and dimes. She has a goal of $1. Suppose she collects 7 coins. a. Write an equation that relates q and d to her goal of $1. b. Write an equation that relates q and d to the 7 coins that she collected. d = # of dimes c. Graph both equations on the same grid. (q, d) Use an interval of 1 on both aes. Calendars q = #of quarters These charts will help ou find the and intercepts. Value Equation: # of Dimes # of Items Equation: # of Quarters d. State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. (Include both values and what it means to both conditions.) Homework is continued

24 ~~ Unit 4, Page 24 ~~ 4. Student s in Eric s gm class must cover a distance of 1,6 meters b running or walking. Most students run part of the wa and walk part of the wa. Eric can run at an average speed of 2 meters per minute and walk an average of 8 meters per minute. He will spend a total of 14 minutes eercising. (time spent running =, time spent walking = ) a. Write an equation that relates the time Eric spends running and walking to his goal of covering 1,6 meters. b. Write an equation that relates and to Eric s total time. c. Graph both equations on the same grid. Use an interval of 2 on the -ais and 2 on the ais. These charts will help ou find the and intercepts. Distance Equation: Walking Minutes Time Equation: Running Minutes d. State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. (Include both values and what it means to both conditions.)

25 ~~ Unit 4, Page 25 ~~ Use graphic methods to solve each sstem. In each case, substitute the solution values into the equations to see if our solution is correct. 1) 2) 3) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) -intercept: (, ) Solution: Solution: Solution: Check Equation 1: Check Equation 1: Check Equation 1: Check Equation 2: Check Equation 2: Check Equation 2: Homework is continued

26 ~~ Unit 4, Page 26 ~~ On our own; #4 (Refer to page 12, #5 for help.) Kallie s Work-outs Kallie is conditioning for tr-outs. She has alread run 1 miles. She will run 2 miles per da. Blake s Work-outs Blake is conditioning for tr-outs. He has alread run 15 miles. He will run 1.5 miles per da. = distance and = #das = distance and = #das Equation: Equation: 2 2 Graph both equations on the following grid. Use an interval of 2 on the -ais and 5 on the -ais. Conditioning for Tr-outs 1) Name the point of intersection: Total Distance 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if the eercise more das and what it means if the eercise fewer das.) On our own; #5 Number of Das (Refer to page 12, #6 for help.) Bethan s Cabin Bethan is renting a cabin in Tennessee. The charge a $2 cleaning fee and $1 per night. = total cost and = #nights Equation: 8 Mand s Hotel Mand is renting a hotel room at the Lodge in Tennessee. The don t charge a cleaning fee and $15 per night. = total cost and = #nights Equation: 6 Graph both equations on the following grid. Use an interval of 1 on the -ais and 1 on the -ais. Staing in TN 1) Name the point of intersection: Total Cost 2) What does the point of intersection mean to the situation? (Include what each value means, what it means if the sta more nights and what it means if the sta fewer nights.) Number of Nights

27 ~~ Unit 4, Page 27 ~~ More Graphs of Linear Sstems (Standard Form: ) I can create a math model for a real life situation using sstem of equations in standard form and a graph. We are going to revisit some situations where ou have alread written the equations. You can refer back to our previous assignments to help ou. Eample 1: You are running a concession stand at the basketball game. You sell hotdogs for $1 and sodas for $2. You sold a total of 12 items. At the end of the night, ou made $2. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Hotdogs, Sodas) Use an interval of 2 on the -ais and 2 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. Eample 2: Beaumont is sponsoring a pancake dinner to raise mone for a field trip. Each adult ticket will cost $2 and each child s ticket will cost $1. You estimate a total of 7tickets to be sold. At the end of the night, ou made $9. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Adults, Children) Use an interval of 1 on the -ais and 1 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation.

28 ~~ Unit 4, Page 28 ~~ Your turn. 1) A test has multiple choice questions worth 2 points apiece and short answer questions worth 4 points apiece. There are a total of 3 questions. The test is worth a total of 1 points. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (MC Questions, SA Questions) Use an interval of 5 on the -ais and 5 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. 2) Justin has saved five dollar bills and singles. Justin has a total of 35 bills. His savings are worth a total of $75. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Fives, Singles) Use an interval of 5 on the -ais and 1 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. Homework is continued

29 ~~ Unit 4, Page 29 ~~ 3) Claire bought sandwiches and drinks at the ballgame. The sandwiches cost $4 each and the drinks were $2 each. Claire bought 9 items for a total of $28. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Sandwiches, Drinks) Use an interval of 1 on the -ais and 2 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. 4) The store at which Michael usuall shops is having a sale. Roast beef costs $4 a pound and shrimp costs $1 a pound. He bought 16 pounds of meat for a total cost of $1. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Roastbeef, shrimp) Use an interval of 2.5 on the -ais and 2 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. Homework is continued

30 ~~ Unit 4, Page 3 ~~ 5) It will take 2 points to make the plaoffs, the hocke team coach told the plaers. We get 2 points for a win and 1 point for a tie. The team has 12 games left in the season. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Wins, Ties) Use an interval of 2 on the -ais and 2 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. 6) You are in charge of buing the hamburger and chicken for a part. You have $6 to spend. The hamburger costs $2 per pound and chicken is $3 per pound. You bought 25 pounds of meat. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Hamburger, Chicken) Use an interval of 5 on the -ais and 5 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation.

31 #1 ~~ Unit 4, Page 31 ~~ Review Writing Sstems of Equations & Solving b Graphing Heather's Bunn Heather has a bunn that weighs 5 pounds and gains 3 pounds per ear. Heather's Cat Heather has a cat that weighs 15 pounds and gains 1 pound per ear. = weight and = time Equation: Equation: 3 7 Graph both equations on the following grid. Use an interval of 1 on the -ais and 2 on the -ais. 1) Name the point of intersection: 2) What does the point of intersection mean to the situation? #2 Fertilizer A You are testing two fertilizers on palm trees. Palm tree A is 8 cm tall growing at a rate of 6cm/da. Fertilizer B You are testing two fertilizers on palm trees. Palm tree B is 2 cm tall growing at a rate of 4cm/da. = height = # of das Equation: Equation: 4 Graph both equations on the following grid. Use an interval of 1 on the -ais and 4 on the -ais. 1) Name the point of intersection: 4 2) What does the point of intersection mean to the situation? Homework is continued

32 ~~ Unit 4, Page 32 ~~ #3 DISH TV Jonathan is getting Dish TV installed. It costs $2 for the installation and $3 per month for the channels he wants. Cable TV Anthon is getting cable TV. There is no installation fee and he will have to pa $5 per month for the channels he wants. = cost = # of months Equation: Equation: 5 5 Graph both equations on the following grid. Use an interval of 1 on the -ais and 5 on the -ais. 1) Name the point of intersection: 2) What does the point of intersection mean to the situation? #4 A class of 27 students went on a field trip. The took 8 vehicles, some buses and vans. Find the number of buses and the number of vans the took if each bus holds 45 students and each van holds 15students. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Buses, Vans) Use an interval of 1 on the -ais and 1.5 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. Homework is continued

33 ~~ Unit 4, Page 33 ~~ #5 Colin has saved quarters and dimes. Colin has a total of 5 coins. He has $8. in his pigg bank. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Quarters, Dimes) Use an interval of 5 on the -ais and 1 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. #6 The Lakers scored a total of 96 points in a basketball game against the Bulls. The Lakers made a total of 4 two-point and three-point baskets. How man two-point shots did the Lakers make? How man three-point shots did the Lakers make? Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Two-point Shots, Three-Point Shots) Use an interval of 4 on the -ais and 4 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation.

34 Solving Sstems b Substitution ~~ Unit 4, Page 34 ~~ I can solve a sstem of equations b substitution. Solve this sstem of equations using substitution. Check. 3-2 = 11 = 9 2 The substitution method is used to eliminate one of the variables b replacement when solving a sstem of equations. Think of it as "grabbing" what one variable equals from one equation and "plugging" it into the other equation. Sstems of Equations ma also be referred to as "simultaneous equations". Let's look at an eample using the substitution method: Solve this sstem of equations (and check): 3 2 = 11 = Replace the "" value in the first equation b what "" now equals. Grab the "" value and plug it into the other equation. 3(9 2) 2 = Solve this new equation for "" = = 11-8 = -16 = 2 4. Place this new "" value into either of the ORIGINAL equations in order to solve for "". Pick the easier one to work with! + 2 = 9 or = 9 2 = 9 2(2) = 9 4 = 5

35 ~~ Unit 4, Page 35 ~~ Solving Sstems b Substitution NOTES 1) 2) Solution: Check solutions Solution: Check solutions 3) Solving Sstems b Substitution Solution: Check solutions 1) Substitute to make one equation with one variable. 2) Solve the equation b UNDOING the order of operations. 3) Substitute our solution back in for our known variable to calculate the second value. 4) Write our solution as a coordinate point. 5) Check our solution b substituting our solution back into both equations.

36 ~~ Unit 4, Page 36 ~~ Solving Sstems b Substitution and Review b Graphing Solve the following sstems of equations using substitution. Check our solutions. 1) 2) 3) Solution: Solution: Solution: Check Equation 1: Check Equation 1: Check Equation 1: Check Equation 2: Check Equation 2: Check Equation 2: Solve the following sstems of equations using substitution. You do NOT have to check our solutions. 4) 5) 6) Solution: Solution: Solution: Homework is continued

37 ~~ Unit 4, Page 37 ~~ 7) 8) 9) Solution: Solution: Solution: Solve the following sstems of equations b graphing. Check our solutions 1) Equations: 11) Equations: 12) Equations: b = ; m = b = ; m = b = ; m = b = ; m = b = ; m = b = ; m = Solution: Solution: Solution: Check Eq. 1: Check Eq.1: Check Eq. 1: Check Eq. 2: Check Eq.2: Check Eq. 2:

38 Solving Sstems b Substitution II ~~ Unit 4, Page 38 ~~ Eamples and NOTES I can solve a sstem of equations b substitution. 1) 2) Step1 Step 1 Step 2 Step 2 Step 3 Step3 Step 4 Step 4 Step 5 Step 5 3) 4) Steps in Solving Sstems b Substitution 1) Substitute to make one equation with one variable. 2) Solve the equation b UNDOING the order of operations. (Isolate the variable.) 3) Substitute our solution back in for our known variable to calculate the second value. 4) Write our solution as a coordinate point. 5) Check our solution b substituting our solution back into both equations.

39 ~~ Unit 4, Page 39 ~~ Solving Sstems b Substitution II and Review of Graphing Solve the following sstems of equations using substitution. Don't forget to find the solution for both variables. Put a rectangle around our solution. 1) 2) 3) 4) 5) 6) Homework is continued

40 ~~ Unit 4, Page 4 ~~ 7) 8) 9) Solve the following sstems of equations b graphing. 1) 11) 12) b = ; m = b = ; m = -intercept: (, ) -intercept: (, ) b = ; m = b = ; m = -intercept: (, ) -intercept: (, ) Solution: Solution: Solution:

41 ~~ Unit 4, Page 41 ~~ I can solve a sstem of equations b elimination. Solving Sstems Using Elimination (also called Addition Method or Combination Method) The addition method of solving sstems of equations is also called the method of elimination. This method is similar to the method ou probabl learned for solving simple equations. If ou had the equation " + 6 = 11", ou would write " 6" under either side of the equation, and then ou'd "add down" to get " = 5" as the solution. + 6 = = 5 You'll do something similar with the addition method. Solve the following sstem using addition. 2 + = 9 3 = 16 Note that, if I add down, the 's will cancel out. So I'll draw an "equals" bar under the sstem, and add down: 2 + = 9 3 = 16 5 = 25 Now I can divide through to solve for = 5, and then back-solve, using either of the original equations, to find the value of. The first equation has smaller numbers, so I'll back-solve in that one: 2(5) + = = 9 = 1 Then the solution is (, ) = (5, 1). It doesn't matter which equation ou use for the backsolving; ou'll get the same answer either wa. If I'd used the second equation, I'd have gotten: 3(5) = = 16 = 1 = 1...which is the same result as before.

42 ~~ Unit 4, Page 42 ~~ Solving Sstems b Elimination NOTES 1) 2) Solution: Solution: 3) Solving Sstems b Elimination 1) Make sure that when ou add our equations, one of the variables will be eliminated. 2) Add the two equations. 3) Solve for the variable. (Isolate) 4) Substitute our solution back in for our known variable to calculate the second value. 5) Write our solution as a coordinate point. 6) Check our solution b substituting our solution back into both equations. Solution:

43 ~~ Unit 4, Page 43 ~~ Solving Sstems b Elimination Solve the following sstems of equations using elimination. Make sure ou find the value of both of the variables. 1) 2) 3) 4) 5) 6) Decide whether to use substitution or elimination method to solve. Solve each sstem. 7) 8) 9)

44 Review Solving Sstems of Equations Use the graphing method to solve each sstem. ~~ Unit 4, Page 44 ~~ 1) 2) 3) Solution: Solution: Solution: Use the substitution method to solve each sstem. 4) 5) 6) Solution: Solution: Solution: 7) Solution: 8) Solution: Homework is continued

45 ~~ Unit 4, Page 45 ~~ Use the elimination method to solve each sstem. 9) 1) 11) Solution: Solution: Solution: 12) 13) 14) Solution: Solution: Solution: Homework is continued

46 ~~ Unit 4, Page 46 ~~ 15) You are buing $48 worth of lawn seed that consists of two tpes of seed. One tpe is a quickgrowing re grass that costs $4 per pound, and the other tpe is a higher-qualit seed that costs $6 per pound. You need a total of 11 pounds of seed. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Quick, Higher Qualit) Use an interval of 1 on the -ais and 1 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation. 16) Your grandmother made 24 oz. of jell. You have two tpes of jars. The smaller holds 1 oz. And the larger holds 12 oz. Your grandmother wants to fi 22 jars. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Small, Large) Use an interval of 2 on the -ais and 2 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation.

47 ~~ Unit 4, Page 47 ~~ 17) You are buing $3 worth of birdseed that consists of two tpes of seed. Thistle seed attracts finches and costs $2 per pound. Dark oil sunflower seed attracts man kinds of sunbirds and costs $1.5 per pound You are buing 18 pounds of birdseed. Define our variables: Write a sstem of equations: Find the -intercept and intercept for both equations. Eq. 1: and Eq 2: and Graph our sstem on the same coordinate grid. (Thistle, Dark) Use an interval of 3 on the -ais and 3 on the -ais) State the coordinates of intersection. Eplain what these coordinates tell ou about the situation.