Section 3.1 Solving Linear Sstems b Graphing Name: Period: Objective(s): Solve a sstem of linear equations in two variables using graphing. Essential Question: Eplain how to tell from a graph of a sstem of linear equations if it will have one solution, no solution, or infinitel man solutions. Homework: Assignment 3.1. #1 15 in the homework packet. Notes: Vocabular A sstem of equations in two variables and consists of two equations. A solution of a sstem of two equations in two variables is an ordered pair that makes both equations true. Checking to See if an Ordered Pair is a Solution 1. Substitute the ordered pair into both equations.. If the ordered pair is a solution to both equations, then it is a solution to the sstem of equations. Determine whether the ordered pair is a solution of the sstem of linear equations. Eample 1:,, 4 Eample : 1, 3, 4 1 4 10 Solving a Sstem of Equations b Graphing 1. Graph both equations of the same set of aes.. Find the point of intersection. Reflection: 1
E: Solve the sstem b graphing: 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,. Solve the sstem b graphing. Eample 3: 3 4 Eample 4: 1 Eample 5: 6 8 3 7 Eample 6: 3 6 6 Reflection:
Special Cases If the graphs of the equations in a sstem are parallel (do not intersect), then the sstem has NO SOLUTION. Sstems with no solution are called inconsistent. If the graphs of the equations in a sstem are the same line (coincident), then the sstem has INFINITELY MANY SOLUTIONS. Sstems with infinite solutions are called dependent. Sstems with solutions are called consistent. Tell if the sstem is inconsistent, the equations are dependent, or consistent. Eample 7: 8 3 6 4 Eample 8: 6 3 1 8 Besides graphing, what is another wa to tell if lines are parallel? Parallel lines have the slope and -intercepts. How can ou tell the slope of a line if the equation is in standard form? Does the sstem have one solution, no solution, or an infinite number of solutions? Eample 9: 4 16 1 1 3 4 4 Eample 10: 6 6 14 Reflection: 3
Section 3. Solving Linear Sstems Algebraicall Objective(s): Solve a sstem of linear equations in two variables using substitution and linear combinations. Solve application problems involving sstems of equations. Essential Question: When choosing a method for solving a sstem of equations, when would ou use linear combinations, and when would ou use substitution? Homework: Assignment 3.. #16 3 in the homework packet. Notes: Solving a Sstem of Linear Equations b Substitution: E: Solve the sstem 3 5 7 4 8 b substitution. Step One: Since and -4 8 are EQUAL, one can replace the other. Substitute -4 8 into the first equation replacing. Solve. Step Two: Substitute the value from Step One into either of the original equations and solve for the remaining variable. Step Three: Write our answer as an ordered pair and check in both of the original equations. 3 4 8 5 7 1 4 5 7 17 51 3 4 3 8 4 3 5 7 3 4 5 3 1 15 7 4 8 4 4 3 4 1 8 Solution: 4, 3 Solve the sstem of equations. Eample 1: 10 4 Eample : 5 4 6 56 Reflection: 4
Some sstems of equations do not have an equation that can be solved nicel for one of the variables. If this occurs, we can solve the sstem using a new method. Solving a Sstem of Equations b Linear Combinations (Elimination): E: Solve the sstem 5 3 1 4 6 10 b linear combinations. Step One: Write the two equations in standard form. (These two are alread in standard form) 5 3 1 4 6 10 Step Two: Multipl one or both of the equations b a constant to obtain coefficients that are opposites for one of the variables. We can multipl the first equation b to obtain a -coefficient of 6 in the first equation (the opposite of 6 ) 5 3 1 10 6 4 6 10 1 4 6 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. 6 0 1 6 1 5 3 1 10 3 1 3 9 3 Step Five: Write our answer as an ordered pair and check in the original sstem. Solution:,3 5 3 1 5 3 3 10 9 1 4 6 10 4 6 3 8 18 10 Reflection: 5
Solve the sstem of equations. Eample 3: 4 3 37 3 47 Eample 4: 4 3 4 5 48 Multipl the equation b Eample 5: 3 Eample 6: 5 3 9 5 3 Multipl the equation b Multipl the first equation b Multipl the second equation b Choosing an Appropriate Method: Substitution is the method of choice when one of the equations is easil solvable (or alread solved) for one of the variables. If this is not the case, use linear combinations to solve the sstem. E: Which method would be BEST for solving the following sstem of equations? a) 3 7 0 11 10 5 Linear Combinations b) 4 5 Substitution Reflection: 6
Special Cases: As we know from solving sstems of equations b graphing, sstems of equations can have eactl one solution, infinitel man solutions, or no solution. Sstem of Equations with Infinitel Man Solutions: E: Solve the sstem 9 6 0 1 8 0 using the method of our choice. Note: Because the equations are in standard form, and are not easil solvable for one of the variables, we will use linear combinations. Step One: Done. The equations are in standard form. Step Two: Multipl the first equation b 4 and the second equation b terms. 3 to eliminate the 9 6 0 4 36 4 0 1 8 0 3 36 4 0 Step Three: 0 0 0 0 0 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 the equation has INFINITELY MANY SOLUTIONS. a), then Note: If we were to graph these two equations, the two lines would be the same line. Solve the sstem of equations. Eample 7: 7 3 6 14 Reflection: 7
Sstems of Equations with No Solution: E: Solve the sstem 6 using the method of our choice. 4 5 Note: Because the first equation is easil solvable for, we will use substitution. Step One: 6 6 Step Two: 4 6 5 4 4 1 5 Note: The variable was eliminated! 1 5 If the variable is eliminated, and we end up with a false statement (i.e. a equation has NO SOLUTION. b), then the Note: If we were to graph these two equations, the two lines would be parallel. Solve the sstem of equations. Eample 8: 6 6 9 54 3 Solve b an method. Eample 9: 6 15 Eample 10: 6 3 Reflection: 8
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 sstem of equations. Step Five: Answer the question asked and label the answer appropriatel. Application Problems with Sstems of Equations E: A sporting goods store receives a shipment of 14 golf bags. The shipment includes two tpes of bags, full-size and collapsible. The full-size bags cost $38.50 each. The collapsible bags cost $.50 each. The bill for the shipment is $3430. How man of each tpe of golf bag are in the shipment? Step One: (# of Full-Size Bags) + (# of Collapsible Bags) = (Total # of Golf Bags in the Shipment) (Rate) (# of Full-Size Bags) + (Rate) (# of Collapsible Bags) = (Cost of Shipment) Step Two: # of Full-Size Bags = F # of Collapsible Bags = C Total # of Bags = 14 Rate of Full-Size Bags = 38.50 Rate of Collapsible Bags =.50 Cost of Shipment = 3430 Step Three: F C 14 38.5F.5C 3430 F 14 C 38.5 14 C.5C 3430 F 14 C Step Four: We will use substitution. 4774 38.5C.5C 3430 F 14 84 16C 1344 F 40 C 84 Step Five: There are 40 full-size bags and 84 collapsible bags in the shipment. Reflection: 9
Eample 11: A health store wants to make trail mi with raisins and granola. The owner mies granola, which costs $4 per pound, and raisins, which cost $ per pound, together to make 5 lbs of trail mi. How man pounds of raisins should he include if he wants the miture to cost him a total of $80? Step One: pounds of + pounds of = (Rate) ( ) + (Rate) ( ) = Step Two: lbs of granola = G lbs of raisins= R Total lbs of trail mi = Rate of granola = Rate of raisins = Cost of trail mi = Step Three: (equations) Step Four: (solve) Step Five: (answer) Reflection: 10
Sample CCSD Common Eam Practice Question(s): 1. The equations for two lines are given below: 7 9 4 What is the -coordinate of the point of intersection of the two lines? A. B. 1 C. -1 D. -. What is the -coordinate of the solution of the sstem of equations? 4 10 5 3 18 A. 3 B. 1 C. 1 D. 3 3. A coin bank contains onl dimes and nickels. The bank contains 46 coins. When 5 dimes and nickels are removed, the total value of the coins is $3.40. How man nickels did the coin bank start with? A. 1 B. C. 4 D. 34 Reflection: 11
Section 3.3 Graphing and Solving Sstems of Linear Inequalities Objective(s): Graph the solution set of a sstem of linear inequalities. Essential Question: Describe the procedure for solving a sstem of linear inequalities. Homework: Assignment 3.3. #4 9 in the homework packet. Notes: Vocabular A solution of a sstem of linear inequalities is an ordered pair that is a solution of each inequalit in the sstem. A sstem of linear inequalities is a set of two or more linear inequalities Testing if an Ordered Pair is a Solution to a Sstem of Linear Inequalities E: Use the sstem of linear inequalities 4 6 Is (0, -6) a solution? Test the point in both inequalities. It is a solution if and onl if it satisfies both inequalities. 4 6 0 6 4 6 0 6 6 4 true 6 6 false So (0, -6) is NOT a solution. Test if the ordered pair is a solution to the sstem of inequalities. 4 6 Eample 1: (7, 5) Eample : (0, 0) Reflection: 1
Graphing a Sstem of Linear Inequalities E: Graph the sstem 3 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: Lightl shade the appropriate half-planes for each inequalit. Step Four: The solution to the sstem is the overlapping region formed b the shading in Step Three. Shade darkl this region and erase the regions that have NO overlapping. Step Five: Choose a point in the shaded region (not on either line) and test it in the original sstem of inequalities. Choose (0, ). 3 3 0 0 3 3 0 true 6 3 true Graph the solution to the sstem of linear inequalities. Eample 3: 5 Eample 4: 3 1 Reflection: 13
Eample 5: 1 1 3 Eample 6: 3 6 3 4 1 Eample 7: 5 10 4 1 Eample 8: 5 5 15 4 8 Eample 9: 4 4 3 6 Eample 10: 3 4 8 4 16 Reflection: 14
5 5 Eample 11: 3 Eample 1: 3 1 Sample CCSD Common Eam Practice Question(s): 1. Which graph shows the solution to the sstem of inequalities below? 3 1 3 Reflection: 15
Section 3.4 Linear Programming Objective(s): Solve application problems involving linear programming techniques. Essential Question: Are the vertices of a feasible region the onl possible points that satisf an objective function? Eplain our answer. Homework: Assignment 3.4. #30 36 in the homework packet. Notes: Vocabular A solution of a sstem of linear inequalities is an ordered pair that is a solution of each inequalit in the sstem. The graph of the sstem of constraints in linear programming is called the feasible region. Linear programming is the process of optimizing a linear objective function subject to a sstem of linear inequalities. Solving a Linear Programming Problem E: Find the minimum and maimum value of the function P = + 3 subject to the constraints 0 7 Note: In a linear programming problem, the maimum or minimum alwas occurs at one of the vertices of the feasible region. Step One: Graph the feasible region. (Use the constraints.) Step Two: Find the coordinates of the vertices b solving 3 sstems of equations. 0 0 7 and 7 The vertices are (0, 7), (0, ), and (5, ). Reflection: 16
Step Three: Evaluate the objective function for each of the vertices. P = + 3 0 7 P = (0) + 3(7) = 1 0 P = (0) + 3() = 6 5 P = (5) + 3() = 16 Solution: The maimum value of the function P is 1. It occurs when = 0 and = 7. The minimum value of the function P is 6. It occurs when = 0 and =. Eample 1: A compan makes S pairs of skis and B snowboards under the following constraints: S B 0 8 S B 40 S B 3 Find the maimum profit for the compan if the sell the skis for $70 per pair and the snowboards for $50 each. Note: The horizontal ais is the S-ais, and the vertical ais is the B-ais. Step One: Graph the feasible region. Step Two: Find the coordinates of the vertices b solving 6 sstems of equations. S 0 S 0 S 0 B 8 S B 40 S B 3 (, ) (, ) (, ) B 8 B 8 S B 40 S B 40 S B 3 S B 3 (, ) (, ) (, ) Reflection: 17
Step Three: Evaluate the objective function for each of the vertices. S B P = Solution: The maimum value of the function P is. It occurs when S = and B =. Eample : T C T A compan makes T tape plaers and C CD plaers under the following constraints: 60 100 C 00 Find the maimum profit if the compan sells the tape plaers for $8 each and the CD plaers for $33 each. Step One: Graph the feasible region. Reflection: 18
Step Two: Find the coordinates of the vertices b solving 3 sstems of equations. T 60 T 60 C 100 C 100 T C 00 T C 00 (, ) (, ) (, ) Step Three: Evaluate the objective function for each of the vertices. T C P = Solution: The maimum value of the function P is. It occurs when T = and C =. Sample CCSD Common Eam 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 maimum 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 man of each tpe of vehicle can be parked in the lot to maimize the amount of mone collected? A. 0 buses and 60 cars B. 10 buses and 50 cars C. 0 buses and 0 cars D. 30 buses and 30 cars Reflection: 19