Mini Lecture 3.1 Systems of Linear Equations in Two Variables

Transcription

1 Mini Lecture. Systems of Linear Equations in Two Variables. Determine whether an ordered pair is a solution of a system of linear equations. 2. Solve systems of equations by graphing.. Solve systems of linear equations by substitution. 4. Solve systems of linear equations by addition. 5. Select the most efficient method for solving a system of linear equations. 6. Identify systems that do not have exactly one ordered-pair solution.. Determine whether (2, ) is a solution of the system. x + 2 y = 4 2 x + y = 2. Solve by graphing x + y = 0 y = x + 2. Solve by the substitution method. a. y = 2 x + 9 b. x + 5y = 2 x 4y = 9 x + 4 y = 4. Solve by the addition method. a. 2 x + y = 9 b. x + 5y = 4 5 x + y = 4 2 x + y = Solve by the method of your choice. Identify inconsistent systems and systems with dependent equations. a. x 2y = b. x y = x 6y = 2 6x + 2y A system of linear equations in two variables represents a pair of lines. There are three possibilities for solutions: a. If two lines intersect at one point, then there is exactly one ordered-pair solution. b. If two lines are parallel, then there is no solution. c. If two lines are identical, then there are infinitely many solutions. All three methods for solving systems of linear equations in two variables will produce the same answer; however, one method will sometimes be more efficient than another. Answers:. no 2. (, ). a. ( 9, 9) b. (, ) 4. a. ( 7, 5) b. (, 2) 5. a. Inconsistent; the lines are parallel, no solution b. {(x, y) x y = } or {(x, y) 6x + 2y } the lines coincide and the system has infinitely many solutions. ML-8

2 Mini Lecture.2 Problem Solving and Business Applications Using Systems of Equations. Solve problems using systems of equations. 2. Use functions to model revenue, cost, and profit, and perform a break even analysis. At JoJo s pet and supply store female gerbils sell for $8 each and male gerbils sell for $5 each. On Saturday 20 gerbils were sold for a total of $9. How many males and how many females were sold on Saturday? 2. To promote the grand opening of Leroy s Toys the store gave away 9000 miniature cars and goofy sunglasses. The cars cost and the sunglasses cost 5 each. Leroy spent a total of $290 on the giveaways. How many of each item did he buy?. The larger of two numbers is equal to three times the smaller. If twice the larger is added to three times the smaller, the sum is 27. Find the numbers. 4. Twice the length of a rectangle is equal to five times its width. The perimeter of the rectangle is 77 inches. Find the dimensions of the rectangle. 5. Cashews cost $.60 per pound and almonds cost $2.70 per pound. For a fundraiser, the volleyball team will be selling bags of mixed nuts. How many pounds of cashews and how many pounds of almonds should the team buy in order to make a 60 pound mixture that will sell for $.00 per pound? 6. How many gallons of 5% alcohol solution and how many gallons of 40% alcohol solution should be mixed to get 20 gallons of a 0% alcohol solution? 7. Joe and Jack inherited $200,000 from their Aunt Lulu. They each decided to put their money into savings accounts for year and then decide how to spend it. Joe s money earned 5% interest and Jack s earned.8%. Together, their money earned $8,608 in interest. How much did each boy inherit? 8. A small airplane can travel 600 miles in 4 hours with the wind. The return trip against the wind takes 5 hours. Find the speed of the plan in still air and the speed of the wind. 9. Since Jane s grandparents enjoy making birdhouses and selling them at local craft shows. They will pay $50 for the booth rental for the weekend. The materials for making each birdhouse cost $8.75. If they are able to sell the birdhouses for $5.00 each, how many will they need to sell to beak even? What if they 5 birdhouses? 40 birdhouses? Answers:. females; 7 males sunglasses; 000 cars. and 9 4. inches wide; 27.5 inches long lbs. cashews 40 lbs. almonds 6. 2 gallons of 40%; 8 gallons of 5% 7. Joe $84,000; Jack $6, mph; 5 mph wind birdhouses to break even; they will lose $56.25; they will make $00. ML-9

3 Mini Lecture. Systems of Linear Equations in Three Variables. Verify the solution of a system of linear equations in three variables. 2. Solve systems of linear equations in three variables.. Identify inconsistent and dependent systems. 4. Solve problems using systems in three variables. Examples. Show that the ordered triple (, 2, ) is a solution of the system: x + y + z 2 x y + z x + 2 y z = 4 2. Solve the system: x + y + z = 4 x y z = 2 2 x + 2y z = 2. Solve the system: x + 4y z = 2 2 x 8y + 6z = x y + z = 0 4. Solve the system: 2 x + y z 4 x + 6y 2z = 0 x 4y + z 5. Create three equations from the stated problem and then solve. The sum of the three numbers is 4. The largest is 4 times the smallest, while the sum of the smallest and twice the largest is 8. A system of linear equations is three variables represents three planes. A linear system that intersects at one point is called a consistent system and has an ordered triple as an answer (x, y, z). A linear system that intersects at infinitely many points is also called a consistent system and is also called dependent. A linear system that has no common point(s) of intersection represents an inconsistent system and has no solution. Answers:. +2+=6, 2 2+=, +4 9= 4 2. (,,2). No Solution, Inconsistent system. 4. infinitely many solutions, dependant equations 5. x + y + z = 4, z = 4x, x + 2 z = 8. The numbers are 2, 4 and 8. ML-20

4 Mini Lecture.4 Matrix Solutions to Linear Systems. Write the augmented matrix for a linear system. 2. Perform matrix row operations.. Use matrices to solve linear systems in two variables. 4. Use matrices to solve linear systems in the variables. 5. Use matrices to identify inconsistent and dependent systems.. Write the matrix for each system. a. x 4y = 2 b. 2 x + y = 7 c. x + 2 y + z = 0 x + 6 y 4 x = 8 x 4y + z 2 x + y 5z = 0 2. Write the system of linear equations represented by each augmented matrix. a. 4 6 b. c Perform each matrix row operation as indicated and write the new matrix a. b. 2 4 c R R 2 2R + R2 R R2 and 2R + R2 4. Solve each system using matrices. a. x + 0y = b. x + y + x c. 2 x + 2y + z = x + 2 y = x + y z x + y + 2z x + 2 y 2z = 9 x + 2 y + 4z = 0 Organization and neatness is very important when using matrices to solve systems. Caution students to watch signs carefully. Using the calculator is a great way to check systems solved using matrices. Students tend to panic if fractions happen. Encourage them to keep working through the problem x y = Answers:. a. b. 6 5 c a. 2x + y = x + y + z 2x y b. c. x + y 2z =. a. x + y = 0 b. x + 2y + 2z = c a. (, ) b. (, 2, 2) c. ( 2,, ) ML-2

5 Mini Lecture.5 Determinants and Cramer s Rule. Evaluate a second-order determinant. 2. Solve a system of linear equations in two variables using Cramer s rule.. Evaluate a third order determinant. 4. Solve a system of linear equations in three variables using Cramer s rule. 5. Use determinants to identify inconsistent and dependent systems.. Evaluate the determinant of each of the following matrices: 0 8 a. b Use Cramer s rule to solve the system: 2 x y 4 x 2y = 0. Evaluate the determinant of the following matrix: Use Cramer s rule to solve the system: x + y + z x + y + 2 z = 7 2 x + y + z = 4 5. Use Cramer s rule to solve each system or to determine that the system is inconsistent or contains dependent equations. a. 2 x 4y b. 2 x + 6y x + 2y x + y = 8 A matrix of order m n has m rows and n columns. a b a b The determinant of a 2 x 2 matrix is denoted by a2 b = a b2 a2b. 2 a2 b2 If the determinant D = 0 and at least one of the determinants in the numerator is not 0, then the system is inconsistent and there is no solution. If the determinant, D = 0 and all the determinants in the numerators are 0, then the equations in the systems are dependent and the system has infinitely many solutions. Answers:. a. b (, ) (2,, ) 5. a. inconsistent b. dependent equations; infinitely many solutions ML-22