Lesson Title



Name/Period: _______________________________________Unit icHomeworkHW √On-TimeJan. 4 (A)Jan. 5 (B)7.1Introduction to SystemsSolving Systems by GraphingPage 431: 8-23Essential Question: How can we determine the solution of a linear system given a graph?Jan. 6 (A)Jan. 7 (B)7.3Solving Systems Using Elimination, part IPage 447: 3-33 odd, 39-41Essential Question: How can we determine how many solutions a linear system has?Jan. 8 (A)Jan. 11 (B)7.4Quiz #1 – Graphing & EliminationSolving Systems Using Elimination, part IIPage 454: 3-20Essential Question: How can we use the graphing calculator to verify algebraic solutions?Jan. 12 (A)Jan. 13 (B)7.27.5Solving Systems Using SubstitutionSolving Special Linear SystemsPage 439: 3-23 odd; 31, 32, 35, 36Page 463: 8-23 oddEssential Question: How do you determine which method of solving a linear system is most efficient?Jan. 14 (A)Jan. 15 (B)7.6Quiz #2 – Elimination & SubstitutionSystems of Linear InequalitiesPage 469: 3-23Essential Question: How can we determine the solution of a system of linear inequalities given a graph?Jan. 19 (A)Jan. 20 (B)Unit ReviewPage 479: #1-30 & Study for TestEssential Question: How do you determine which method of solving a linear system is most efficient?Jan. 21 (A)Jan. 22 (B)Unit 7 TestEssential Question: What could I have done to better prepare for the exam?Solving Systems by GraphingTwo or more linear equations form a system of linear equations. Any ordered pair that makes all of the equations in a system true is a solution of a system of linear equations.The solution is where the systems are equal.The equations in a system of linear equations use the same variables.The solution to a linear system is the point(s) that make both equations true.The solution to a linear system is the intersection of the graphs of the linear equations.Example:The lines intersect at (1, 3).Steps to Solve a Linear Equation by GraphingWrite each equation in slope-intercept form. (y = mx + b)Carefully graph each line.Plot the y-intercept (b).Use the slope (m) to plot the second point.Connect the dots to create the lines (don’t forget the arrows!).Find the point where the lines intersect and write it as an ordered pair.Check your solution by substitution (manually or with the graphing calculator).Examples: Solve by graphing.x+y=42x-y=5x+y=03x-2y=10y= -53x+3y=13x-3Finding the Solution to a Linear System on a Graphing CalculatorY =Enter the equations in the screen.GRAPHGraph the equations using the feature.CALCUse the feature. Choose INTERSECT to find the point where the lines intersect. Note: Press ENTER 3 times to display the intersection.Example: Find the intersection of the system using the graphing calculator.y= -5x+6y= -x-2Practice. Solve each system using the graphing calculator, identify the solution, and sketch the graph.y=-12x-1y=14x-4-4x+y=3x= -y-2Word Problem Practice. Create the linear system equations and graph to identify the solution.Find the value of two numbers if their sum is 12 and their difference is 4. THS is selling tickets to a choral performance. On the first day of ticket sales there were 3 senior citizen tickets sold and 1 child ticket sold for a total of $19. On the second day, THS made $26 by selling 3 senior citizen tickets and 2 child tickets. Find the price of each type of ticket – senior citizen and child.Solving Systems Using EliminationIn the elimination method, you use the Addition and Subtraction Properties of Equality to add or subtract equations in order to eliminate a variable in the system. The elimination method is also known as linear combination.We use linear combinations to eliminate one of the variables so that we can solve for the other one.Then, we can substitute the value that we found for the variable for which we solved into one of the original equations and solve for the other variable.Example. Find the solution to the system using elimination.2x+5y=176x-5y= -9Step 1:Eliminate one variable. Since the sum of the y coefficients is 0, add the equations to eliminate y.2x+5y=176x-5y= -9---------------------8x+0=8Add the two equations.x=1Solve for xStep 2: Substitute 1 for x to solve for the eliminated variable, y.2x+5y=17You can use either equation. We are using the first.21+ 5y=17Substitute 1 for x.2+5y=17Simplifyy=3Solve for y.Since x = 1 and y = 3, the solution is (1, 3).Steps for Solving Systems Using Elimination/Linear CombinationArrange the equations with like terms in columns.Look for coefficients that are opposites for one of the variables. For example, 2x and -2x are opposites (they eliminate each other when added together).Note: If you have no opposites, you may have to multiply one, or both, of the equations to create an opposite.For example: If one equation has 2x and the other has 4x, multiple the equation with 2x by -2 to create the opposite (-4x). Remember to multiply the entire equation to keep the equation true.Add the two equations together to eliminate one of the variables.Solve for the variable and substitute it into one of the original equations.Solve for the second variable.Check your solutions (through substitution in the second original equation) and then write your answer as an ordered pair.Examples. Solve the linear systems using elimination/linear combination.-x+2y= -811x+3y=1x+6y= -16-5x-3y=-7Sometimes you have to rearrange the equations to get the terms in the same order.-3x-2y= -811x= -3y+12y=12-5x-5x-3y= -7Sometimes you have to multiply one of the equations to get opposite coefficients.3x+6y=125x-2y=3-x+3y=65x+y= -9Practice. Solve the linear systems using elimination/linear combination.-4x-2y= -12x-y=114x+8y= -242x+y=19-6x+6y=6-7x+y=-19-6x+3y= -12-2x+3y=-19The senior classes at High School A and High School B planned separate trips to New York City. High School A rented and filled 1 van and 6 busses with 372 students. High School B rented and filled 4 vans and 12 busses with 780 students. Each van and bus carried the same number of students. How many students can a van carry? How many students can a bus carry?Elimination/Linear Combination – Modifying BOTH EquationsSometimes, you need to multiply BOTH equations in order to get coefficients that are opposite.Example. Solve the linear system.2x+2y=-8There is no opposite to eliminate. Need to multiply3x-3y=18both equations to create an opposite.32x+2y=-8 (3)6/-6 can be created as an “opposite” for y by multiplying23x-3y=18 (2)the top equation by 3 and the bottom by 2.6x+6y= -246x-6y=36--------------------12x=12x=12x+2y= -821+ 2y= -82+2y= -82y= -10y= -5The solution set is (1, -5).Practice.2x-5y= -195x+4y=-303x+2y=03x-9y=-18Solving Systems by SubstitutionYou can solve linear systems by solving one of the equations for one of the variables. Then, substitute the expression for the variable into the other equation. This is called the substitution method.Using substitution. What is the solution to the system (use substitution).y=3xx+y= -32Because y = 3x, you can substitute 3x for y in the second equation.x+3x= -324x= -32x= -8y=3(-8)y= -24The solution is (-8, -24).Oftentimes, you will need to solve for a variable in order to use the substitution method. You may not be given a solved equation like the one above, y = 3x. You will need to use your properties of equality to solve for a variable in one equation and then substitute the expression into the second equation.Example. What is the solution to the system? Use substitution to solve.2y+4x=24-2x+y= -4Solve for a variable in one equation:-2x+y= -4+2x +2x-------------------------y=2x-4Substitute the solution into the other equation.22x-4+ 4x=244x?8+4x=248x+8=24 -8 -8---------------------8x=16x=2Substitute the solution into the first equation to solve for the other variable.-22+ y= -4-4+y= -4+4 +4----------------------y=0The solution set is 2, 0).y=6x-11-7x-2y=-13-2x-3y=-7x-2y=11Practice. Use substitution to solve the system.2x-3y=-1-3x-3y=3y=x-1y= -5x-17y=5x-7-4x+y=6-3x-2y=-12-5x-y=21-7x-2y= -13-5x+y=-2x-2y=113x-8y=24Practical Applications of Systems (Real World Scenarios)You can solve systems of linear equations using a graph, substitution, or elimination/linear combination. The best method depends on the forms of the given equations.Choosing a Method for Solving Linear SystemsMethodWhen to useGraphingWhen you want a visual display or when you want to estimate a solution.SubstitutionWhen one equation is already solved for one of the variables, or when it is easy to solve for one of the variables.Elimination/Linear CombinationWhen the coefficients of one variable are the same or opposites, or when it is not convenient to use graphing or substitution.Examples.Finding a Break-Even Point (i.e. when income = expenses)A fashion designer makes and sells hats. The material for each hat costs $5.50. The hats sell for $12.50 each. The designer spends $1,400 on advertising. How many hats must the designer sell to break even?Step 1. Write a system of equations. Let x = the number of hats sold, and let y = the dollars of expense or income.Expense: y = 5.5x + 1400Income: y = 12.5xStep 2. Choose a method to solve. Substitution is recommended because both equations are already solved for y.476821573025 y=5.5x+1400Write the first equation 12.5x=5.5x+1400Substitute7x=1400Subtract 5.5x from both sidesx=200Divide both sides by 7.Since x is the number of hats, the designer must sell 200 hats to break even.A puzzle expert wrote a new Sudoku puzzle book. His initial costs are $864. Binding and packaging each book costs $0.80. The price of the book is $2.00. How many copies must be sold to break even?Solving a Mixture ProblemA dairy owner produces low fat milt containing 1% fat and whole milk containing 3.5% fat. How many gallons of each type should be combined to make 100 gallons of milk that is 2% fat?Let x = the number of gallons of low fat milk, and let y = the number of gallons of whole milk.Total gallons: x+ y = 100Fat content: 0.01x + 0.035y = 0.02(100)The first equation is easy to solve for x or y so use substitution.x+y=100Write the equationx=100-ySubtract y from both sides to solve for x.Substitute 100 – y for x in the second equation and solve for y.0.01x+0.035y=0.02(100)Write the first equation.0.01100-y+ 0.035y=0.02(100)Substitute1-.01y+0.035y=2Distribute1+0.025y=2Simplify0.025y=1Subtract 1 from both sidesy=40Divide both sides by 0.025Substitute y = 40 in either equation and solve for x.x+40=100Substitutex=60Subtract 40 from both sidesThe owner should mix 60 gallons of low-fat milk with 40 gallons of whole milk.One antifreeze solution is 20% alcohol. Another antifreeze solution is 12% alcohol. How many liters of each solution should be combined to make 15 liters of antifreeze solution that is 18%?Practice.Gym A charges a membership fee of $100 plus $2.00 per visit. Gym B charges a membership fee of $60, plus $4.00 per visit. At how many visits, x, is the total cost, y, the same for both gyms?A ticket booth sold 226 tickets and collected 843 in ticket sales. Adult tickets are $5.50 and child tickets are $1.50. How many tickets of each type were sold?Solving Systems of InequalitiesA system of inequalities is made up of two or more linear inequalities. A solution of a system of linear inequalities is an ordered pair that makes all the inequalities in the system true. The graph of a system of linear inequalities is the set of points that represent all of the solutions to the system.Each point on a dashed line is not a solution. A dashed line is used for inequalities with > or <.Each point on a solid line is not a solution. A solid line is used for inequalities with > or <.The overlap of the shaded areas (including the solid line) represents the solution set. y > xy < -xExamples. Graph each system.y≤1y>2x-4x≥2y>-13x+2y≤3x+1y>23x-3x+y>2y≤23x+1Practice. Graph each system.y≤-x-2y≤-3y≥-5x+2y<53x+2y≤12x+2y<-2x-3Identifying Solutions to Linear InequalitiesUse substitution to determine if an ordered pair is a solution to a given system. If the ordered pair makes both inequalities true, then it is a solution. Otherwise, it is not a solution.Practice. Determine whether the ordered pair is a solution of the given system.(-2, -4)(1, 1)y≤-52x- 23x+2y ≥-2y<-12x+2x+2y≤2Graphing Linear Inequalities with a Graphing CalculatorA graphing calculator can show the solutions of an inequality or a system of inequalities. To enter an inequality, press APPS and scroll down to select Inequalz. Move the cursor over the = for one of the equations. Notice the inequality symbols at the bottom of the screen. They correspond to the buttons below labeled F2, F3, F4, and F5 in green. Change the = symbol to the appropriate inequality symbol by pressing ALPHA followed by the corresponding F2-F5 button.Move the cursor to the right, using your arrow, and enter the expression for Y1.Press enter.Move the cursor to the left and change the = symbol to the appropriate inequality symbol by pressing ALPHA followed by the corresponding F2-F5 button.Move the cursor to the right, using your arrow, and enter the expression for Y1.Press enter.Press GRAPH to graph the system of inequalities.Example. Use a graphing calculator to graph the system of inequalities. Sketch the graph.y<-2x-3y≥x+4 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download