Lesson 6: Using Matrices to Solve Systems of Equations

Transcription

1 Opening Exercise In Module 1, ou learned to solve sstems of equations b graphing, and b the substitution and elimination methods. Sstems of equations can also be solved using matrices. Before we use matrices, let s use a simple example to review solving sstems of equations. 1. Solve the sstem b graphing and then use either the substitution method or the elimination 3x+ = 5 method to verif our results. Graphing Method Substitution or Elimination Method 3x+ = 5 3x+ = 5 The solution to 3x+ = 5 is x = and =. Unit 4+: Networks & Matrices S.49

2 To solve the sstem,, with matrices, we have to see this sstem as three matrices as shown 3x+ = 5 below. 1 2 x 4 = 5 2. What does each matrix represent? 3. Determine 1 2 x. How does it relate to the original equation? x 1 2 To solve for the matrix, we need to use some tpe of operation to move the matrix to the other side of the equation. In algebra we use inverse operations to solve equations and we ll do the same 1 2 here. We need the inverse of the matrix. When ou multipl a matrix, A, b its inverse, A -1, ou get the identif matrix, I. AA -1 = A -1 A = I Suppose A, X and B are matrices and AX = B, then AA -1 X = A -1 B. This gives I X = A -1 B or X = A -1 B Unit 4+: Networks & Matrices S.50

3 To determine the inverse, A -1, ou ll need a special number called the determinant, and written as A or det A. If a b A = c d, then the determinant of A or det A is given b the formula A = ad bc. 4. Find the determinant of the matrix 1 2. To find the inverse ou ll use the reciprocal of the inverse and then manipulate the matrix. The steps are given below. If a b A = c d and A = ad bc, then 1 1 d b A = A c a. 5. If the determinant of the matrix is 0, then no inverse exists. Wh is that so? 6. What is the inverse of matrix A? Unit 4+: Networks & Matrices S.51

4 7. Solve the sstem using A x 4 = 5 Write A -1 here. And here. 1 2 x 4 = 5 8. Does our answer agree with our work in Exercise 1? 9. Which method, graphing, substitution, elimination or matrices do ou like best? Wh? Unit 4+: Networks & Matrices S.52

5 We ve been looking at transportation problems throughout this unit, but most transportation problems are so complex that ou need computers or graphing calculators to solve them. Most require at least 4 x 4 matrices. We ll look at some tpical 2-variable sstem problems which will give us 2 x 2 matrices. 10. Admission to the count fair is $2 for children and $5 for adults. Last Sunda, 1900 people came to the fair and the collected $5900 at the entrance booth. A. If C represents the number of children who attended and A represents the number of adults, what are the two equations to represent this situation? B. Write the matrix equation that represents this sstem. C. Find the determinant and the inverse, and then solve the sstem with matrices. D. Number of children: Number of adults: Unit 4+: Networks & Matrices S.53

6 11. The sum of two numbers is 1 and their difference is 15. A. Write a sstem of equations for the two numbers. Be sure to define the variables ou are using. B. Solve the sstem using matrices. Unit 4+: Networks & Matrices S.54

7 Lesson Summar You can use matrices to solve sstems of equations. Complete the example below b following the steps given. Steps Explained Example: Solve 2 x = 8 x+ 2= 7 using matrices 1. Write the matrix equation. 2. Find the determinant. 3. Determine the inverse matrix. 4. Use the inverse to isolate the variable matrix. 5. Multipl to solve for the variables. 6. Check our answer. Unit 4+: Networks & Matrices S.55

8 Homework Problem Set 1. Solve the sstem, x 2= 1 x+ 4= 8 using matrices. In Exercise 5, ou explained wh matrices with a determinant of 0 has no inverse. Find the determinant of each matrix and then decide which have no inverse Julie went to the Taco Truck and bought 5 tacos and 2 burritos for $ Kent bought 3 tacos and 4 burritos for $ Use matrices to determine how much each taco costs. Unit 4+: Networks & Matrices S.56

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