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1 Syllabus Objective: 3.1 The student will solve systems of linear equations in two or three variables using graphing, substitution, and linear combinations. System of Two Linear Equations: a set of two linear equations with two variables Solution of a System of Two Linear Equations: an ordered pair x, y that satisfies both equations in the system; the point at which the two lines intersect Checking if an ordered pair is a solution to a system of two linear equations: Ex: Is the ordered pair 2, 2 a solution of the system 2x y 2? x y 4 Step One: Substitute the ordered pair for x, y in both equations. 2x y , 4 is a solution of the first equation. x y , 4 is NOT a solution of the second equation. Step Two: If the ordered pair is a solution to both equations, then it is a solution of the system. So, NO, 3, 4 is not a solution of the system. Solving a System of Two Linear Equations by Graphing: Ex: Solve the system by graphing: 2 x y 2 x y 4 Step One: Graph both equations on the same coordinate plane. Step Two: Find the coordinates of the point of intersection of the two lines. The lines appear to intersect at the point 2, 2. Step Three: Substitute the ordered pair found in Step Two into the original equations of the system to determine if this point is the solution of the system of equations. 2x y , 2 is a solution of the first equation. x y , 2 is a solution of the second equation. So, 2, 2 is the solution of the system of equations. Page 1 of 24

2 Special Cases: If the graphs of the equations in a system are parallel (do not intersect), then the system has NO SOLUTION. Systems with no solution are called inconsistent. Systems with solutions are called consistent. If the graphs of the equations in a system are the same line (coincident), then the system has INFINITELY MANY SOLUTIONS. Systems with infinite solutions are called dependent. Note: If the graphs are not coincident and intersect, then the system has EXACTLY ONE SOLUTION. Consistent solution(s) Inconsistent Ø Dependent ( ) Independent (1 or Ø) Ex: How many solutions does each system of equations have? a) 6x2y 8 3x y 7 b) x2y 2 3x6y 6 Step One: Graph both equations on the same coordinate plane. a) b) b) Step Two: Using the graph, determine the number of solutions. a) The lines are parallel, therefore there is no solution to the system. The system of equations is inconsistent. b) The lines are coincident (the same line), therefore there are infinitely many solutions to the system. Page 2 of 24

3 Solving a Linear System on the Graphing Calculator: Ex: Solve the linear system 6 x 9 y 13 x2y 10 on the graphing calculator. Step One: Solve both equations for y. (Rewrite in slope-intercept form.) 6x9y 13 9y 6x y x 3 9 x2y 10 2y x10 1 y x5 2 Step Two: Type the two equations into Y1 and Y2. Step Three: Graph the equations. Step Four: Use the Intersect command to find the point of intersection. The Intersect command is under the Calc menu. Note: 6 was the guess. You may enter in a value for the guess or arrow over to the point. Approximate Solution: 5.524,2.238 You Try: Solve the system by graphing. x 2y 6 y 3x4 QOD: Explain how to tell from a graph of a system of linear equations if it will have one solution, no solution, or infinitely many solutions. Write a unique example of each. Page 3 of 24

4 Syllabus Objective: 3.1 The student will solve systems of linear equations in two or three variables using graphing, substitution, and linear combinations. Solving a System of Linear Equations by Substitution: Ex: Solve the system 3 x 5 y 27 x4y 8 by substitution. Step One: Solve one of the equations for one of the variables (if necessary). Note: You may choose which variable to solve for. Step Two: Substitute the expression from Step One into the other equation of the system and solve for the other variable. Step Three: Substitute the value from Step Two into the equation from Step One and solve for the remaining variable. Step Four: Write your answer as an ordered pair and check in both of the original equations. x 4y 8 x 4y8 3 4y8 5y y24 5y y 51 y 3 x x 4 3x5y x4y Solution: 4, 3 Ex: Solve the system 2 x 6 y 15 by substitution. Use the steps listed above. 2 y x Step One: This step is complete. The second equation is solved for x. Step Two: 2 2y 6y 15 4y6y 15 10y 15 3 y 2 Step Three: x 2 y 3 x Step Four: 3 2x6y y x2 3 2 Solution: 3 3, 2 Page 4 of 24

5 You Try: Solve the system 2 x y 6 3x2y 2 pair into both of the original equations. by substitution. Check your answer by substituting the ordered Some systems of equations do not have an equation that can be solved nicely for one of the variables. If this occurs, we can solve the system using a new method. Solving a System of Equations by Linear Combinations (Elimination): Ex: Solve the system 5 x 3 y 1 4x6y 10 Step One: Write the two equations in standard form. by linear combinations. 5x 3y 1 4x 6y 10 Step Two: Multiply one or both of the equations by a constant to obtain coefficients that are opposites for one of the variables. We can multiply the first equation by 2 to obtain a y-coefficient of 6 in the first equation (the opposite of 6 ) 5x 3y x6y 2 4x 6y x6y 10 Step Three: Add the two equations from Step Two. One of the variable terms should be eliminated. Solve for the remaining variable. Step Four: Substitute the value from Step Three into either one of the original equations to solve for the other variable. Step Five: Write your answer as an ordered pair and check in the original system. 6x0y 12 6x 12 x 2 523y1 10 3y 1 3y 9 y 3 Solution: 2,3 5x3y x 6y Page 5 of 24

6 Ex: Solve the system 5 x 3 y 9 by linear combinations. Use the steps listed above. 2x5y 23 Step One: This step is complete. The equations are written in standard form. Step Two: Eliminate the x term by multiplying the first equation by 2 and the second equation by 5. (Note: You could have also eliminated the y term by multiplying the first equation by 5 and the second equation by 3.) 5x3y 9210x6y 18 2x5y x25y 115 Step Three: 19 y 133 y 7 Step Four: 5x 3y 9 5x x x 30 x 6 Step Five: 5x3y x5y Solution: 6, 7 15x2 y 31 You Try: Solve the system 4x6y 11 by linear combinations. Choosing an Appropriate Method: Substitution is the method of choice when one of the equations is easily solvable (or already solved) for one of the variables. If this is not the case, use linear combinations to solve the system. Ex: Which method would be BEST for solving the following system of equations? a) 3x7y 20 11x10y 5 Linear Combinations b) y x 4 y 2x 5 Substitution Note: For extra practice, solve each system of equations using the method you chose. Page 6 of 24

7 Special Cases: As we know from solving systems of equations by graphing, systems of equations can have exactly one solution, infinitely many solutions, or no solution. System of Equations with Infinitely Many Solutions: Ex: Solve the system 9 x 6 y 0 12x8y 0 using the method of your choice. Note: Because the equations are in standard form, and are not easily solvable for one of the variables, we will use linear combinations. Step One: Done. The equations are in standard form. Step Two: Multiply the first equation by 4 and the second equation by 3 to eliminate the y terms. 9x6y x24 y 0 12x8y x24 y 0 Step Three: 0 x 0 y Note: Both of the variables were eliminated! If both of the variables are eliminated, and we end up with a true statement (i.e. a a), then the equation has INFINITELY MANY SOLUTIONS. Note: If we were to graph these two equations, the two lines would be coincident. Systems of Equations with No Solution: Ex: Solve the system 2x y 6 4x2y 5 using the method of your choice. Note: Because the first equation is easily solvable for y, we will use substitution. Step One: 2x y 6 y 2x 6 Step Two: 4x2 2x6 5 4x4x Note: The variable was eliminated! If the variable is eliminated, and we end up with a false statement (i.e. a b), then the equation has NO SOLUTION. Note: If we were to graph these two equations, the two lines would be parallel. Page 7 of 24

8 QOD: When choosing a method for solving a system of equations, when would you use linear combinations, and when would you use substitution? Page 8 of 24

9 Syllabus Objective 3.3 The student will solve application problems involving systems of equations/inequalities. When solving an application problem, it is helpful to have a problem solving plan. We will use the following plan to solve the application problems that follow. Problem-Solving Plan: Step One: Write a verbal model. Step Two: Assign labels. Step Three: Write an algebraic model. Step Four: Solve the algebraic model using one of the methods for solving a system of equations. Step Five: Answer the question asked and label the answer appropriately. Application Problems with Systems of Equations Ex: A sporting goods store receives a shipment of 124 golf bags. The shipment includes two types of bags, full-size and collapsible. The full-size bags cost $38.50 each. The collapsible bags cost $22.50 each. The bill for the shipment is $3430. How many of each type of golf bag are in the shipment? Step One: (# of Full-Size Bags) + (# of Collapsible Bags) = (Total # of Golf Bags in the Shipment) (Cost of Full-Size Bags) (# of Full-Size Bags) + (Cost of Collapsible Bags) (# of Collapsible Bags) = (Cost of Shipment) Step Two: # of Full-Size Bags = F # of Collapsible Bags = C Total # of Bags = 124 Cost of Full-Size Bags = Cost of Collapsible Bags = Cost of Shipment = 3430 Step Three: F C F 22.5C 3430 Step Four: We will use substitution. F 124 C C 22.5C C22.5C C 1344 C 84 F F F 124 C Step Five: There are 40 full-size bags and 84 collapsible bags in the shipment. Page 9 of 24

10 Ex: Sheila and a friend share the driving on a 280 mile trip. Sheila s average speed is 58 miles per hour. Her friend s average speed is 53 miles per hour. Sheila drives one hour longer than her friend. How many hours did each of them drive? Step One: We will use the equation d rt (distance equals rate times time). (Sheila s Distance) + (Friend s Distance) = 280 (Sheila s Distance) = (Sheila s Average Rate) (Sheila s Time) (Friend s Distance) = (Friend s Average Rate) (Friend s Time) (Sheila s Time) = 1 + (Friend s Time) Step Two: Sheila s Distance = d S Friend s Distance = d F Sheila s Avg Speed = 58 Friend s Avg Speed = 53 Sheila s Time = t S Friend s Time = t F Step Three: d d 280 S F d d S F 58t S 53t F t S 1 t F Step Four: We will use substitution tF 53tF tS 53tF t F tF53tF 280 t 2 Step Five: Sheila drove 3 hours, and her friend drove 2 hours. F t t S S 1 t F 123 Ex: A chemist needs to make 200L of a 62% solution by mixing together an 80% solution with a 30% solution. How much of each solution should she use? Step One: (Amount of 80% Sol.) + (Amount of 30% Sol.) = (Total Amount of 62% Sol.) (% of Sol.) (Amount of 80% Sol.) + (% of Sol.) (Amount of 30% Sol.) = (% of Sol.) (Amount of 62% Sol.) Step Two: Amount of 80% Sol. = E Amount of 30% Sol. = T Amount of 62% Sol. = 200 % of Sol. We will write all percents as decimals: 62% = 0.62, 80% = 0.8, 30% = 0.3 Step Three: ET E0.3T E T E0.3T 124 Step Four: We will use linear combinations. (Multiply the first equation by 0.3 to eliminate the T term.) E T E0.3T E T E0.3T 124 E 128 T 72 Step Five: The chemist should use 128L of the 80% solution and 72L of the 30% solution. Page 10 of 24

11 Ex: A plane flies 1800 km each way of a two-way trip. The trip takes 3 hours with the wind and 3 hours, 20 minutes against the wind. What is the speed of the plane and the wind in mi/hr? Step One: Distance with the Wind = Distance Against the Wind = 1800 Distance With the Wind = (Speed of Plane + Speed of Wind) Time With the Wind in Minutes Distance Against the Wind = (Speed of Plane Speed of Wind) Time Against the Wind in Minutes Step Two: Speed of Plane = P Speed of Wind = W Time With the Wind in Minutes = 180 Time Against the Wind in Minutes = 200 Step Three: 1800 PW P180W PW P200W 1800 Note: To make it easier to solve later, we can divide out the GCF in both equations. PW PW 10 9 Step Four: We will solve using linear combinations. Multiply the first equation by 5 and the second equation by 3 to eliminate the P term. PW PW 2P P PW W 10 W 0.5 Step Five: Note The solutions above are measured in miles/minute. We want the answers in miles/hour. mi 60 min mi P min 1 hr hr mi 60 min mi W min 1 hr hr Solution: The speed of the plane is 570 mi/hr, and the speed of the wind is 30 mi/hr. You Try: 1. The sum of two numbers is 82. One number is 12 less than 3 times the other. Find the numbers. 2. A health store wants to make trail mix with raisins and granola. The owner mixes granola, which costs $4 per pound, and raisins, which cost $2 per pound, together to make 25 lbs of trail mix. How many pounds of raisins should he include if he wants the mixture to cost him a total of $80? QOD: Write a unique application problem that involves systems of equations. Page 11 of 24

12 Sample CCSD Common Exam Practice Question(s): 1. The equations for two distinct lines are given below: y 7x9 y 4x2 What is the x-coordinate of the point of intersection of the two lines? A. 2 B. 1 C. 1 D What is the x-coordinate of the point of intersection for the two lines below? 4x 2y 10 5x3y 18 A. 3 B. 1 C. 1 D A coin bank contains only dimes and nickels. The bank contains 46 coins. When 5 dimes and 2 nickels are removed, the total value of the coins is $3.40. How many nickels did the coin bank start with? A. 12 B. 22 C. 24 D. 34 Sample SAT Question(s): Taken from College Board online practice problems. 1. Tom and Alison are both salespeople. Tom s weekly compensation consists of $300 plus 20 percent of his sales. Alison s weekly compensation consists of $200 plus 25 percent of her sales. If they both had the same amount of sales and the same compensation for a particular week, what was that compensation, in dollars? Grid-In Page 12 of 24

13 2. At the beginning of 2006, both Alan and Dave were taller than Boris, and Boris was taller than Charles. During the year, Alan grew 2 inches, Boris and Dave each grew 4 inches, and Charles grew 3 inches. Of the following, which could NOT have been true at the beginning of 2007? (A) Alan was shorter than Boris. (B) Alan was shorter than Charles. (C) Boris was shorter than Dave. (D) Dave was shorter than Alan. (E) Dave was shorter than Charles. 3. A convenience store sells small bottles of juice for $2 each and large bottles of juice for $3 each. Ciara bought 8 bottles of juice at this store and paid $18. Some of the bottles of juice that she bought were large, and the rest were small. How many small bottles of juice did Ciara buy? (A) Two (B) Three (C) Four (D) Five (E) Six 4. A 19-liter mixture consists by volume of 1 part juice to 18 parts water. If x liters of juice and y liters of water are added to this mixture to make a 54-liter mixture consisting by volume of 1 part juice to 2 parts water, what is the value of x? (A) 17 (B) 18 (C) 27 (D) 35 (E) 36 Page 13 of 24

14 Syllabus Objectives: 3.2 The student will graph the solution set of a system of linear inequalities. 3.3 The student will solve application problems involving systems of equations/inequalities. System of Linear Inequalities: a set of two or more linear inequalities Solution of a System of Linear Inequalities: all of the ordered pairs that satisfy all of the linear inequalities in the system Testing if an Ordered Pair is a Solution to a System of Linear Inequalities Ex: Use the system of linear inequalities 2 x y 4. Is the given point a solution? y 2x 6 a) 0,0 b) 7,5 c) 0, 6 Test each point in both inequalities. It is a solution if and only if it satisfies both inequalities. 2x y a) 04false 2x y b) 94true y 2x true 2x y 4 c) true y 2x false a) So 0,0is NOT a solution. b) So 7,5 IS a solution. c) So 0, 6 Graphing a System of Linear Inequalities IS NOT a solution Ex: Graph the system y22 x1. x3y 3 Step One: Graph each line on the same coordinate plane. Step Two: Determine whether to use solid or dashed lines. (Recall: Use solid lines for and, and use dashed lines for < and >.) Step Three: Lightly shade the appropriate half-planes for each inequality. Step Four: The solution to the system is the overlapping region formed by the shading in Step Three. Shade only this region. Step Five: Choose a point in the shaded region (not on either line) and test it in the original system of inequalities. Choose 0,2. y2 2x true x 3y true Page 14 of 24

15 Ex: Write a system of inequalities for each graph. a) b) Step One: Write the equations of the lines. a) 1 y x 1, y x 1 b) y x3, y x 1 3 Step Two: Write the inequalities for each line. a) 1 y x 1, y x 1 b) y x3, y x 1 3 Application Problems Involving Systems of Inequalities: Use the problem-solving plan. Ex: A contractor needs at least 500 bricks and at least 10 bags of sand. Bricks weigh 2 lb each and sand weighs 50 lb per bag. The maximum weight that can be delivered is 3000 lb. Write and graph a system of inequalities that represents the situation. Step One: # of Bricks 500 # of Bags of Sand 10 (Weight of Bricks) (# of Bricks) + (Weight of Sand) (# of Bags of Sand) 3000 Step Two: # of Bricks = B # of Bags of Sand = S Weight of Bricks = 2 Weight of Sand = 50 Step Three: B 500 S 10 2B 50S 3000 Step Four: Graph (Note: The horizontal axis is the B-axis, and the vertical axis is the S-axis.) Page 15 of 24

16 Sample CCSD Common Exam Practice Question(s): Which graph shows the solution to the system of inequalities below? x 3y 12 3x y 2 Page 16 of 24

17 Syllabus Objective: 3.4 The student will solve application problems using linear programming techniques. Linear Programming: the process of optimizing a linear function (called the objective function) based upon a system of linear inequalities, which are called constraints. Feasible Region: the graph of a system of constraints Optimization: the process of finding the minimum or maximum value of a quantity Note: In a linear programming problem, the maximum or minimum always occurs at one of the vertices of the feasible region. Solving a Linear Programming Problem Ex: Find the minimum and maximum value of the function P 2x 3y subject to the x 0 constraints y 2. x y 7 Step One: Graph the feasible region. (Use the constraints.) Step Two: Find the coordinates of the vertices. The vertices are 0,7, 0,2, and 5,2. Step Three: Evaluate the objective function for each of the vertices. 0,7 : P ,2 : P , 2 : P Solution: The maximum value of the function P is 21. It occurs when x 0 and y 7. The minimum value of the function P is 6. It occurs when x 0 and y 2. Page 17 of 24

18 Ex: It takes a company 2 hours to manufacture a pair of skis and 1 hour to manufacture a snowboard. The finishing time for both skis and snowboards is 1 hour. The maximum time available for manufacturing is 40 hours and for finishing is 32 hours each week. The manufacturer must produce at least 8 snowboards every week. The profit for a pair of skis is $70, and the profit for a snowboard is $50. Write an objective function for profit and use linear programming to find the maximum profit. How many skis and snowboards should the company manufacture to maximize the profit? Step One: Write the objective function (what is being maximized/minimized). P 70S 50B (S = # of skis, B = # of snowboards) Step Two: Write and graph the constraints. Manufacturing Time: 2S B 40 Finishing Time: S B 32 Number of Skis: S 0 Number of Snowboards: B 8 Note: The horizontal axis is the S-axis, and the vertical axis is the B-axis. Step Three: Find the coordinates of the vertices of the feasible region. The vertices are 0,8, 0,32, 16,8, and 8,24. Step Four: Evaluate the objective function for the vertices to find the maximum profit. P P P P 0,8 : ,32 : ,8 : ,24 : Therefore, the maximum profit is $1760. The company should manufacture 8 skis and 24 snowboards each week to maximize the profit. Page 18 of 24

19 You Try: A company makes tape players (for a profit of $28 each) and CD players (for a profit of $33 each). The company wants to produce at least 60 tape players and 100 CD players per day, but can't produce more than 200 total combined. Write an objective function for profit and use linear programming to find the maximum profit. How many tape players and CD players should the company manufacture to maximize the profit? QOD: Are the vertices of a feasible region the only possible points that satisfy an objective function? Explain your answer. Page 19 of 24

20 Sample CCSD Common Exam Practice Question(s): The area of a parking lot is 600 square meters. A car requires 6 square meters and a bus requires 30 square meters of space. The lot can handle a maximum of 60 vehicles. Let b represent the number of buses and c represent the number of cars. The diagram below represents the feasible region based on the constraints of the number of vehicles that can be parked in the lot. To park in the lot, a bus costs $8 and a car costs $3. How many of each type of vehicle can be parked in the lot to maximize the amount of money collected? A. 0 buses and 60 cars B. 10 buses and 50 cars C. 20 buses and 0 cars D. 30 buses and 30 cars Page 20 of 24

21 Syllabus Objective: 3.1 The student will solve systems of linear equations in two or three variables using graphing, substitution, and linear combinations. System of Three Linear Equations: a set of linear equations with three variables Solution of a System of Three Linear Equations: the ordered triple x, yz, that satisfies all three equations in the system Solving a System of Three Linear Equations (using substitution) Ex: Solve the system. 5x2yz 7 x2y2z 0 3yz 17 Step One: Solve one of the equations for one of the variables. (Solve the third equation for z.) 3yz 17 z 17 3y Step Two: Substitute the expression from Step One into both of the other equations. 5x2y 173y 7 5x2y173y 7 5x5y 10 x 2y2 173y 0 x2y346y 0 x8y 34 Step Three: Solve the system of two equations and two variables created in Step Two using substitution (or linear combinations). x8y 34 x 8y34 5x 5y 10 58y34 5y 10 40y170 5y 10 45y 180 y 4 x 8y 34 x 8434 x 2 Step Four: Find the third variable by substituting the values from Step Three into the equation from Step One. z 17 3y z Step Five: Write your answer as an ordered triple, and check it in all three original equations. Solution: 2,4,5 Page 21 of 24

22 Solving a System of Three Linear Equations (using linear combinations) Ex: Solve the system. x5yz 16 3x3y2z 12 2x4yz 20 Step One: Choose two of the equations and eliminate one of the variables using linear combinations. x5yz 1633x15y3z 48 3x3y2z 1213x3y2z 12 18y 5z 36 Step Two: Choose one of the equations from Step One and the leftover equation to eliminate the same variable using linear combinations. x5yz 1622x10y2z 32 2x4yz 2012x4yz 20 6y 3z 12 Step Three: Use the resulting equations from Steps One and Two. Solve for the remaining variables in this system of two equations using linear combinations. 18y5z y5z 36 6y3z y9z 36 4z 0 z 0 6y 3z 12 6y 12 6y y 2 Step Four: Find the third variable by substituting the values from Step Three into any of the three original equations. x5yz 16 x1016 x x 6 Step Five: Write your answer as an ordered triple, and check it in all three original equations. Solution: 6,2,0 Page 22 of 24

23 Application Problem Involving Systems of Three Linear Equations Ex: Determine the quadratic function that contains the points 1,3, 2,10, and 2, 6. 2 Step One: Use the standard form of the equation of a quadratic function: y ax bx c. Substitute each point into the equation for x and y to obtain a system of equations with three variables, a, b, and c. 2 3a 1 b 1 c 3 abc 2 10 a 2 b 2 c 10 4a2bc 2 6a 2 b 2 c 64a2bc Step Two: Solve the system of three linear equations using the method of your choice. (Shown below is the linear combination method.) abc 31abc 3 4a2bc 10 4a2bc 10 abc 31abc 3 4a2bc 64a2bc 6 3ab713ab 7 3a3b 93a3b 9 3a b 7 3a 3b 9 4b 16 b 4 3a b 7 3a 4 7 a 1 abc 3 c 2 You Try: 1 4 c 3 Solution: 2 y x x 4 2 Substituting,, and into 2 a b c y ax bx c a) Solve the system using the method of your choice. x 2yz 4 2x yz 4 x 3yz 7 b) Yuki borrows a total of $10,000 from three different banks at three different interest rates, 18%, 15%, and 9%. She borrows 3 times as much at 15% as she borrows at 18%. The total interest is $ How much did she borrow at each rate? (Use your calculator for computation.) QOD: Which method of solving a system of equations do you prefer to use, and why? Page 23 of 24

24 Sample CCSD Common Exam Practice Question(s): What is the x-coordinate of the solution to the system of equations? 2x y z 5 x 3z 14 2x 3y 2z 2 A. 1 B. 3 C. 4 D. 5 Sample SAT Question(s): Taken from College Board online practice problems. 3x2y2z 19 3x yz If the equations above are true, which of the following is the value of y z? (A) 5 (B) 4 (C) 0 (D) 4 (E) 5 2. The sum of r and p is equal to twice s, and p is 36 less than twice the sum of r and s. What is the value of r? Grid-In Page 24 of 24

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