Name: Date: Period: CHAPTER : LINEAR SYSTEMS AND THEIR GRAPHS Notes #: Section.: Solving Linear Sstems b Substitution The solution to a sstem of equations represents the where the. It would be great if we could find a wa to do this without having to graph the lines!! A. Substitution Method If in function notation, rewrite as x s and s. (The function part will be our ) Choose one equation - get alone Substitute ( ) this variable with the new expression in the second equation USE PARENTHESES!! Solve Substitute back in the first equation to solve for the other variable Express our answer as a point (, ) in alphabetical order Solve using the substitution method:.) = - x + =.) b = a b = a.) = -x + x + =.) m n = m + n = -,
.) d = c d = c +.) C(n) = -n C(n) = n + Possible strange answers:.) x =.) + x = x = x + = B. Word Problems in variables Write a let statement (often x = st number, = nd number) Translate into an equation Translate into an equation Solve the sstem of equations using the substitution method Write let statements, translate to two equations, and solve using substitution:.) The sum of two numbers is. The difference of these two numbers is. Find the numbers. 0.) The sum of two numbers is. One number is twelve more than the other. Find the numbers.
.) The sum of two numbers is. One number is three times the other. Find the numbers..) A outh group with members is going to the beach. There will also be five chaperones that will each drive a van or car. Each van seats people, including the driver. Each car seats people, including the driver. How man vans and cars will be needed? You can now solve linear sstems (a set of lines) using algebra (substitution/elimination) AND using coordinate Geometr (graphing). You should get the same answer for both methods. - solve the equations using substitution; write our answer as a point (, ) - graph the two lines using either the intercept method OR slope-intercept method - confirm that the two lines intersect (meet) at our solution point.) = x x + = st Method: substitution or elimination nd Method: graphing (graph both lines on the coordinate plane below) 0-0 - - - - - - - - - - 0 x - - - - - - - - -0 solution: (, )
Notes #: Section -: Solving Sstems Using Elimination The solution to a sstem of equations represents the where the. A. Addition/Elimination Method Line up the two equations in Standard Form Choose either the x s or s to cancel ou want the (the number out front) to be equal but sign. If this doesn t match et, one/both equations b a number so that the coefficients do match. Add the equations together watch one variable disappear. Solve and substitute back in the first equation to solve for the other variable Express our answer as a point (, ) Solve using the addition method:.) x = x + =.) x + = 0x =.) c + d = - c d =.) x 0 x 0
Possible strange answers:.) x =.) x + = x = x + = B. Applications: Word Problems.) On a special da, tickets for a minor league baseball game cost $ for adults and $ for students. The attendance that da was, and $0 was collected. Write and solve a sstem of equations to find the number of adults and the number of students that attended the game..) Suppose the band sells cans of popcorn for $ per can and cans of mixed nuts for $ per can. The band sells a total of 0 cans and receives a total of $. Find the number of cans of popcorn and the number of cans of mixed nuts sold.
.) The sum of two numbers is. The difference of the two numbers is. Find the numbers. 0.) The sum of two numbers is. The difference of the two numbers is. Find the numbers. You can now solve linear sstems (a set of lines) using algebra (substitution/elimination) AND using coordinate Geometr (graphing). You should get the same answer for both methods. - solve the equations using elimination; write our answer as a point (, ) - graph the two lines using either the intercept method OR slope-intercept method - confirm that the two lines intersect (meet) at our solution point.) x = - x + = st Method: substitution or elimination solution: (, ) nd Method: graphing (graph both lines on the coordinate plane below) 0-0 - - - - - - - - - - 0 x - - - - - - - - -0
Notes #: Section -: Applications of Linear Sstems **Now that we know how to solve a sstem of equations in variables, we have the tools that we need to solve more complex word problems** Steps:. Write let statements. Write two equations in words.. Replace the words with our variables.. Solve b the substitution or addition method.. Write our answers in words.) A baseball team plaed games. The won more games than the lost. How man games did the lose? Let l = # of games lost w = games lost + games won = total plaed games won = +.) Four pencils and two pens cost $0.. Six pencils and five pens cost $.. Let x = # of (pencils) + (pens) = 0. (pencils) + (pens) =. = # of
.) Rikki has 00 coins, all dimes and quarters. She has more quarters than dimes. How man of each coin does she have? a) Let d be the number of dimes and q be the number of quarters. Write an equation that shows that the sum of the number of dimes and quarters is 00. b) Write an equation that shows that she has more quarters than dimes. c) Re-write these two equations and solve using the substitution or addition method. Answer the problem in a complete sentence. Often, ou will need to solve a word problem that discusses unit rate. This sounds like cents per mile or $.0 per minute etc. These equations are written in a slightl different wa and should alwas be solved using substitution. Total Cost = Flat Fee + Unit Rate( ).) Budget Rent-A-Truck rents small trucks at a dail rate of $. plus 0 cents per mile. Avis Rent-A-Truck rents the same size truck at a dail rate of $. plus cents per mile. For what mileage is the cost the same? Let m = mileage c = total cost (Budget) Cost = Flat Fee + Unit Rate(miles) (Avis) Cost = Flat Fee + Unit Rate(miles)
.) Acme rents a pickup truck at a dail rate of $. plus cents per mile. Speedo Rentzit rents a pickup for $. plus cents per mile. For what mileage is the cost of the same. Let m = c = (Acme) (Rentzit).) A+ Taxi Service charges a flat rate of $.00 plus $.0 per mile. Top Dog Taxi Service charges a $.00 flat rate and $.0 per mile. a) Define two variables and write two equations. b) After how man miles would the cost for each service be the same? c) If ou rode for less than 0 miles which service should ou hire? Wh?.) Suppose ou own a tping service. You bu a personal computer for $0 on which to do our tping. You charge $.0 per page for tping. Expenses are $.0 per page for ink, paper, electricit, and other expenses. How man pages must ou tpe to break even?
.) A chemist has one solution that is 0% acid. She has another solution that is % acid. How man liters of each solution should she combine to get 0 liters of a 0% acid solution? Solution Solution Final Solution.) Suppose it takes ou hours to fl about 00 miles from Miami, Florida to Seattle, Washington. At the same time, our friend flies from Seattle to Miami. His plane travels with the same average airspeed, but his flight onl takes hours. Find the average airspeed of the planes. Find the average wind speed. Miami to Seattle Distance Rate Time Seattle to Miami 0
0.) Jon can row km downstream in hours but it takes him hours to row the same distance upstream. What is the speed of the current and his rate in still water? Upstream Distance Rate Time Downstream.) A plane has a speed of 0kph in still air. It can travel 0km with the wind in the same time that it would take to travel 0km against the wind. Find the speed of the wind. With the wind Distance Rate Time Against the wind
Notes #: Section -: Linear Inequalities Reminder: when graphing these equations. a) x How do we show that x cannot include? b) x How do we show that x can include? Graphing Linear Inequalities: Get alone Graph the line using m and b decisions: Line: If, connect points with a If, connect points with a Shading: If, shade the line If, shade the line Graph each inequalit:.) x 0-0 - - - - - - - - - - 0 x - - - - - - - - -0
.) x 0-0 - - - - - - - - - - 0 x - - - - - - - - -0.) x.) Suppose our budget allows ou to spend no more than $ for decorations for a part. Streamers cost $/roll and tablecloths cost $ each. Use intercepts to graph the inequalit that represents the situation. Find three possible combinations of streamers and tablecloths ou can bu. 0-0 - - - - - - - - - 0 x - - - - - - - - - -0
Section -: Sstems of Linear Inequalities When graphing two lines, the solution to the sstem is. When graphing two linear inequalities (shaded lines), the solution to the sstem is. Solve b graphing; be sure to darken our solution area. Tr color-coding our lines..) + x < > -x +.) + x > x 0 0-0 - - - - - - - - - - 0 x - - - - - - - - -0 0-0 - - - - - - - - - - 0 x - - - - - - - - -0
.) x < x +.) x + - x -.) x + x + < 0-0 - - - - - - - - - - 0 x - - - - - - - - -0 0-0 - - - - - - - - - 0 x - - - - - - - - - -0 0-0 - - - - - - - - - 0 x - - - - - - - - - -0
Notes #: Section.: Negative and Zero Exponents A. Review of Exponents base exponent The represents the number of times the is multiplied b itself. The exponent applies to onl number or smbol unless parenthesis are used. x = ( )( ) = versus (x) = ( )( ) = Evaluate:.).).) -.).) -.) (-).) x if x = - and = - B. Simplifing Zero Exponents Explore: x.) x.) x x 0.) **For ever non-zero number a, a 0 = ** Evaluate:.) 0.) - 0.) x 0.) 0 C. Simplifing Negative Exponents Explore: x.).) x x x.)
**When I have a exponent, I make the exponent and move that term onl to the.** a, a a a - n -n Simplif. Express using positive exponents:.) x -.) m - 0.) -.).) x -.) x a b 0 c Evaluate each expression if: x = and = - Re-write each expression using positive exponents Plug in values of x and. Evaluate..) x -.) - x -.) x - x.).) x 0 -.) x - x
Write a sstem of equations to model each problem and solve..) You have coins that are onl nickels and pennies. The coins total cents. How man of each coin do ou have? a) write let statements to define our variables b) write two equations c) solve.) Claire bought sandwiches and drinks for her friends for $.0. Susan bought sandwiches and drinks for $.00. Find the cost of a sandwich and a drink. a) write let statements to define our variables b) write two equations c) solve.) The sum of two numbers is. The difference of the two numbers is. Find the numbers. a) write let statements to define our variables b) write two equations c) solve.) Joe and Jane each have a savings account. Joe started with $0 and is saving $ a week. Jane started with $ and is saving $ per week. After how man weeks will the have the same amount of mone in their accounts? a) write let statements to define our variables b) write two equations c) solve
Algebra Stud Guide Name: For #-, solve the sstem b graphing. x.) x 0-0 - - - - - - - - - - 0 x - - - - - - - - -0.) x x 0 For #-, solve the sstem using the substitution method. x x.).) x x 0-0 - - - - - - - - - - 0 x - - - - - - - - -0 For #-, (a) define two variables (let statements), (b) write two equations, and (c) solve using substitution..) Trucks-R-Us rents a pickup truck at a dail.) You have $ in an account and ou are rate of $0 plus $0.0 per mile. Trucks--U rents saving $ per week. Your friend has $ in an a pickup truck for $ plus $0. per mile. For account and is saving $ per week. After how what mileage is the cost the same? man weeks will ou both have the same amount of mone in our accounts? (a) (a) (b) (b) (c) (c)
.) The perimeter of a rectangle is cm. The length is cm less than twice the width. Find the dimensions of the rectangle. (a) (b) (c) For #-0, do not solve for x and but state whether the linear sstem will have no solution, one solution, or infinitel man solutions. Explain our reasoning. x x.).) x x.) x x 0.) x x For #-, solve using the addition method. x 0 0.) x.) x x 0.) x 0 x 0. 0
For #-, (a) define two variables (let statements), (b) write two equations, and (c) solve using the addition method..) The sum of two numbers is 0. Their.) Popee goes grocer shopping on Mondas. difference is. Find the numbers. One Monda, he bus gallon of milk and cans of spinach for $.0. The next Monda, he bus (a) gallon of milk and cans of spinach for $.0. What is the cost of a can of spinach? What is the cost of a gallon of milk? (b) (a) (c) (b) (c).) Adam has a collection of coins that are worth $.0. If he onl has nickels and dimes, how man of each coin does he have? (a) (b) (c)
.) Fruit Juice A contains % pure fruit juice and drink B contains % pure fruit juice. How much of each kind of drink should ou mix together to get L of a 0% concentration of fruit juice? Volume % Concentration Amt. Juice Juice A Juice B Mixture For #-, graph each linear inequalit..) x > 0-0 - - - - - - - - - 0 x - - - - - - - - - -0.) x + For #0-, graph each sstem of linear inequalities. 0.) + x <.) + x > - > -x + x 0 0-0 - - - - - - - - - 0 x - - - - - - - - - -0 0-0 - - - - - - - - - - 0 x - - - - - - - - -0 0-0 - - - - - - - - - - 0 x - - - - - - - - -0