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Name: _____________________________________ NOTES: SubstitutionOne method of solving system of equations is by algebraic substitution.Example: Solve x=7-3y and 2x-4y=-6.27-3y-4y=-6Substitute 7-3y for x in the second equation. 14-6y-4y=-6Solve for y-10y=-20y=2Substitute 2 for y in either one of the two original equations to find the value of x.x+32=7x+6=7x=1The solution of this system is (1, 2)Using substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.1. x=32y+x=32. y=22x-4y=1Name: ______________________________________NOTES substitution continued3. y=3x-73x-y=74. y=-x+32y+2x=45. x=2y0.25x+0.5y=10Name: ____________________________________HOMEWORK: Solving Systems of Equations by Substitution1. 4x+5y=14y=23. -7x-5y=21y=x-92. y=2x-2x-2y=244. y=4x-93x-3y=0Name: ____________________________1. y=x+1-4x+3y=-43. -x-7y=20y=-25. y=-3x+9-4x-4y=4Classwork on Substitution2. y=-6x-15-8x+5y=14.8x+3y=-8y=-4x6. y=-5x+123x+2y=-4Name: ______________________________NOTES:Elimination Using Addition & SubtractionIn systems of equations where the coefficient of the x or y terms are additive inverses, solve the system by adding the equations. Because one of the variables is eliminated, this method is called elimination.Example: Use elimination to solve the system of equations.x-3y=7 and 3x+3y=9Add the two equations x-3y=7Substitute 4 for x4-3y=7 3x+3y=9in either original -3y=7-4 4x=16equation and solve -3y=3 x=4for y. y=-1The solution of the system is (4, -1)Use elimination to solve each system of equations. 1. 2x+2y=-22. 4x-2y=-13x-2y=12-4x+4y=-2Name: ______________________________NOTES:Elimination Using Addition & Subtraction continued3. x-y=24. 6x+5y=4x+y=-36x-7y=-205. 2x-3y=124x+3y=24Name: ________________________________HOMEWORK: Elimination Using Addition & Subtraction1. -5x-8y=-23 5x-3y=122. -9x+2y=-7 2x-2y=143.-7x-3y=30-4x-3y=12Name: ______________________________Classwork: Elimination Using + or -Solve each system by elimination.1.7x+2y=192. 5x+3y=-198x-2y=26-3x-3y=153. 9x+2y=84. -3x-7y=178x-2y=26-3x+3y=27Name: _________________________________Classwork continued5. x+4y=206.-7x+2y=-258x+4y=20 -7x+y=-237. 9x-2y=-138. -9x-y=5 -x+2y=5 8x+y=-4-1, 2(1) (3, 4)(2, 1)(2) (-4, 3)2, -1(3) (3, -4)(-2, -1)(4) (-1, 4)Name: _______________________________ NOTES: Elimination Using MultiplicationSome systems of equations cannot be solved simply by adding or subtracting the equations. One or both equations must first be multiplied by a number before the system can be solved by elimination. Consider the following example:Example: Use elimination to solve the following system of equations.x+10y=3 and 4x+5y=5x+10y=3multiply x+10y=3 by -4.-4x-40y=-124x+5y=5Then add the equations. 4x+5y=5 -35y=-7 y=15Substitute 15 for y into either original equation and solve for x.x+1015=3 x+2=3The solution of the system is 1, 15 x=1Use elimination to solve each system of equations.1.3x+2y=02. 2x+3y=6 x-5y=17 x+2y=5Name: _____________________________NOTES: Elimination Using Multiplication continued3. 3x-y=24. 4x+5y=6x+2y=36x-7y=-20Name: _______________________________ NOTES Day 2Elimination Using MultiplicationSome systems of equations cannot be solved by elimination in their original forms. Some equations may have to be multiplied by a number to continue with our elimination. What do we have to multiply the top equation by to eliminate the x's?1. -x+18y=-3 2x-9y=6Some systems of equations have coefficients that are different but also the same sign. In this case, you must first decide what number to multiply to make the coefficients the same, but also make it negative to change the sign. What do we have to multiply the first equation by to eliminate the y's.2.-10x+y=1-2x+3y=-25Name: ______________________________Notes: Elimination = Day 2Solve each system by elimination.3. -16x+8y=-24 8x-7y=-154. 12x-2y=6 -6x-3y=-155. 9x-4y=-30 -7x-y=116. 6x-15y=6 x+5y=-29Name: ____________________________________Homework Day 2 Elimination1. 8x+5y=-10-16x-8y=162. 12x-9y=24[HINT: multiply the bottom equation by something to make the -3 6x-3y=12the opposite of -9.]Name: __________________________________CLASSWORKNotes: Checking solutions to systems. Often you can check to see if an ordered pair is a solution to a system of equations.1. y=-2x-9 -5x+5y=15 (a) -4, -1 (b) -4, 1 (c) (-1, 4) (d) (1, -2)2. y=x+5 -x-4y=-5 (a) 3, 2 (b) -2, 3 (c) (-3, 2) (d) (3, -2)3. -3x-2y=-8 3x+y=10 (a) -2, -2 (b) -4, -2 (c) (4,-2) (d) (-2, 2)4. 9x+4y=8 -8x+8y=16 (a) 9, 0 (b) -2, 0 (c) (-9, 0) (d) (0, 2)Name: __________________________________CLASSWORK continued5. Show that -5,-8 is a solution to the following system:y=2x+2-8x+3y=166. Show that (5 ,7) is a solution to the following system:-8x+3y=-19 -4x+y=-13Name: _______________________________________CLASSWORKWhy Do I Eliminate and How? ActivityCut out the boxes with systems of equations and phrases located at the bottom of the sheet. Look at each system and decide which variable would be easier to eliminate. Place the system under the correct variable on the chart. If changes need to be made to the system, include those changes under the category “How To.”. When completed tape or glue each system and “how to” in the correct column.ELIMINATE xELIMINATE yHOW TOMULTIPLY BY -24x+2y=85x-3y=4x+2y=3 -x+y=-2MULTIPLY BY 3x+4y=72x-4y=-3 2x+5y=6-x+2y=3MULTIPLY BY (2)-2x+3y=7 2x-5y=-3 2x-3y=-23x+3y=4Name: _______________________________ NOTES: Graphing Systems of EquationsTwo or more linear equations involving the same variables form a system of equations. The solution set for the system is the set of ordered pairs that satisfy both equations. One method for solving a system of equations is to graph the equations on the same coordinate plane.127635036639500Example: Solve each system of equations by graphing.x+y=2The point (3, -1) lies on both lines, thusx-y=4(3, -1) is the solution set for the systemof equations.3x+y=2no solutiony=2x+1infinitely many solutions3x+y=42y=4x+2 Use the graphs below to determine whether each system has one solution, no solution, or infinitely many solutions. If the system has one solution, name it.1. x-y=12. x+3y=33.2x+3y=64.x-y=2x+y=-12x+y=22x+3y=-62x-2y=4Name: ______________________________________HOMEWORK:Solve each system of graphing.1. y=-16x+12. 0=3y+5x+6y=-32x-7-9y=-3x+72 Name: ______________________________________HOMEWORK continued3. y=-x-24. 7x-y=-2y=12x+47x-y=4 Name: ________________________________________CLASSWORK3225165464820001. x+y=53x-y=72.2x+y=3-2x+5y=-9Name: ________________________________CLASSWORK continued3140075-17716500 3. 3x+5y=02x-5y=-254. 2x-y=-5-2x-5y=11-904884320858graphing them00graphing themName: ____________________________NOTES: Graphing Systems of Inequalities334327537846000The solution of a system of inequalities is the set of all ordered pairs that satisfy both inequalities. To find the solution of the systemy>x+2y≤-2x-1graph each inequality. The graph of each inequality is called a half-plane. The intersection of the half-planes represents the solution of the system. The graphs of y=x+2 and y=-2x-1 are the boundaries of the region. An inequality containing an absolute value expression can be graphed by graphing an equivalent system of two inequalities.Solve each system of inequalities by graphing.1. y≥2x2. 5x-2y<6y≥-1y>-x+141910010795Name: ____________________________NOTES & HOMEWORK: continued3. y>x4.-x+y≤6y≥-xx+y≤255245057155. Write a system of inequalities for the graph at the right. 444817514033500Name: ____________________________NOTES & HOMEWORK: continued6. x-y≥-17. x-y≤24x-y≥22x+y<1 Name: _____________________________HOMEWORK continued8. 2x-3y≥-39.3x-y<32x+3y<-9x-2y≤-4 Name: ____________________________CLASSWORK: graphing systemsSolve each system by graphing.1. y=-16x+12. x+y=1y=-32x-73x+3y=3 3.y=-78x-84. y=x+2y=14x+1y=2x-1 Name: _____________________________________CLASSWORK continued39052505080005. Graph the solution set to the system of inequalities.2x-y<3 and 4x+3y≥0-2286001041400039814501916430006. Graph the solution set to each system of inequalities. x-y>5x>-17. y≤x+4y≤4-xName: ____________________________________NOTES:Word Problems for Systems of Equations2667000340360001. The Williams family is going to the Johnstown Summer Carnival. They have two ticket options, as shown in the table below.Ticket OptionAdmission Price171767535560Price Per RideA$530?B$380?a) Write an equation that represents the cost per person for each option.b) Graph the equations and estimate a solution. Explain what the solution means.c) Solve the system using substitution.d) Write a short paragraph advising the Williams family which option to choose.Name: ____________________________NOTES on Word Problems of Systems continued2. The sum of two numbers is 18. The sum of the greater number and twice the smaller number is 25. Find the numbers.3. Lina is preparing to take the Scholastic Assessment Test (SAT). She has been taking practice tests for a year, and her scores are steadily improving. She always scores about 150 points higher on the math test than on the verbal test. She needs a combined score of 1270 to get into the college she has chosen. If she assumes she will still have that 150-point difference between the two tests, how high does she need to score on each part?4. Christy has 42 nickels and dimes in all. The total value is $2.80. How many nickels and how many dimes does she have?28136853416300032956502400305. Ms. Foisy’s 1st period class can order up to $90 of free pizzas from Gino’s as a reward for selling the most magazines during the magazine drive. They need to order at least 6 large pizzas in order to serve the entire class. If a pepperoni pizza costs $10.00 and a supreme costs $12.00, how many of each type can they order? List three possible solutions. Name: __________________________________HOMEWORK:Word Problems of Systems of EquationsUse a system of equations to solve each problem.1. Find two numbers whose sum is 64 and whose difference is 42.2. Maria spent a long day working the cash register at Musicville during a sale on CDs. For this sale, all CDs in the store were marked either $12 or $10. Just when she thought she could go home, the store manager gave Maria the job of figuring out how many CDs they had sold at each price, so they could write the total in the store records. Maria doesn’t want to sort through hundreds of sales slips, so she decided on an easier way. The counter at the exit of the store says that 500 people left with CDs (limit one per customer) during the sale, and the cash register contains $5750 from the day’s sales. Maria wrote a system of equations for the number of $10 CDs and the number of $12 CDs.a) What was the system of equations?b) How many CDs were sold at each price?3. Three times one number equals twice a second number. Twice the first number is 3 more than the second number. Find the numbers.Name: __________________________________NOTES: 4-step problem solving processUse the 4-step problem-solving process to solve each problem.1. READThe cost of 3 tacos and 1 juice is $7. The cost of 4 tacos and 2 juices is $10. If t= the cost of a taco and j= the cost of a juice, the scenario can be represented by the following system:3t+j=74t+2j=10What is the cost of one taco and one juice?PLANSOLVECHECK2. READThere are 15 coins in Edith’s purse, which are nickels and dimes only. The total value of the coins in her purse is $1.15. If n=nickels and d= dimes, the scenario can be represented by the following system:n+d=150.05n+0.10d=1.15How many nickels and dimes are in Edith’s purse?PLANSOLVECHECKName: __________________________________WORD PROBLEMS – Day 2Show all your work on a separate sheet of paper!Choose the correct answer.1. 14 school festival tickets were sold to adults and children. A total of $38 was collected from these ticket sales. Adult tickets cost $4 each, and child tickets cost $1 each. The system of linear equations below represents x, the number of adult tickets sold, and y, the number of child tickets sold.x+y=144x+y=38How many tickets were sold?6 adult tickets and 8 child tickets8 adult tickets and 6 child tickets10 adult tickets and 4 child tickets12 adult tickets and 2 child tickets2. The perimeter of an isosceles triangle is 16 inches. The length of its base is 2 times the length of one of its other sides. In the equations below, x represents the length of each of its equal sides, and y represents the length of its base.2x+y=16y=2xWhat are the side lengths of the triangle?3 in., 3 in., 6 in.4 in., 4 in., 8 in.4 in., 8 in., 8 in.6 in., 6 in., 8 in.3. Heidi paid $18 for 7 pairs of socks. She bought wool socks that cost $3 per pair and cotton socks that cost $2 per pair. How many pairs of socks did she buy?(a) 2 pairs of wool and 5 pairs of cotton(b) 3 pairs of wool and 4 pairs of cotton(c) 4 pairs of wool and 3 pairs of cotton(d) 5 pairs of wool and 2 pairs of cotton4. A jar contains only dimes and nickels. The total number of coins in the jar is 15. The total value of the coins is $1.00. How many of each type of coin are in the jar?(a) 5 dimes and 10 nickels(b) 7 dimes and 8 nickels(c) 8 dimes and 7 nickels(d) 5 nickels and 10 dimesName: ____________________________________ continued5. Inside the stables, there are only horses and people. The number of horses and people combined is 9. A boy inside the stables added the number of legs of all the horses and the number of legs of all the people combined. He determined that there are 30 legs in total. How many horses and how many people are in the stables?6. An office manager buys 2 office chairs and 4 file cabinets for $380. Next year, she buys 4 office chairs and 6 file cabinets for $660. What is the cost of each office chair, c? What is the cost of each file cabinet, f?7. Nicole has a total of 31 coins in a jar. There are only dimes and quarters in the jar. The value of the dimes and quarters in the jar is $5.50. How many dimes, d, does she have in the jar? How many quarters, q, does she have in the jar?Name: _____________________________________continued8. (#34 Jan 02) A company manufactures bicycles and skateboards. The company’s daily production of bicycles cannot exceed 10, and its daily production of skateboards must be less than or equal to 12. The combined number of bicycles and skateboards cannot be more than 16. If is the number of bicycles and is the number of skateboards, graph on the accompanying set of axes the region that contains the number of bicycles and skateboards the company can manufacture daily.Name: ___________________________________continued9. A contractor has at most $33 to spend on nails for a project. The contractor needs at least 9 lb of finish nails and at least 12 lb of common nails. How many pounds of each type of nail should the contractor buy?(a) Write a system of three inequalities that describes this situation.(b) Graph the system to show all possible solutions.(c) Name a point that is a solution of the system.(d) Name a point that is not a solution of the system.Name: _________________________________________Review Unit #4Solving Systems of Equations Using the Three MethodsSYSTEM:x+y=2-x+y=-1GRAPHIC Solution: ( , )SUBSTITUTIONELIMINATIONName: _________________________________________Review continued1. Which system of equations has the same solution as the system below?2x+2y=163x-y=4(1) 2x+2y=16 6x-2y=4(2) 2x+2y=166x-2y=8(3) x+y=163x-y=4(4)6x+6y=486x+2y=82. (#4 June ) Given: y+x>2y≤3x-2Which graph shows the solution of the given set of inequalities?Name: __________________________________Review continued3. Suppose you want to spend less that $30 for the meat for subs at a party. Turkey costs $4.00 per pound, and ham costs $6.00 per pound. You want to buy at least 2 pounds of turkey as well as some ham.(a) Write a system of linear inequalities that describes this situation.(b) Graph the system of inequalities.(c) Name three possible solutions.4. You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many of each coin do you have? ................
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