Period+6+-+Applying+Matrices+to+Linear+Systems+(Part+b)


 * APPLYING MATRICES TO LINEAR SYSTEMS (Part b) **

By: Ann Tulley, Kyle Stackpole, & Roger Kim

__**Prerequisite Knowledge:**__


 * Before we apply matrices to linear equations, lets start off with the basics.**


 * Q **. What is a "system of linear equations in two unknowns?"

ax + by = c //where a and b are integers not equal to zero//
 * A. ** First, a **linear equation** in two unknowns x and y is an equation of the form

Remember: To solve a system of linear equations you find the solutions of x and y that satisfies all equations in the system

__**I. Setting up a system of Linear Equations in Matrix Form (Properties): **__



Put the **augmented matrix form** of a single linear equation ax + by = c is just the single row matrix [a b c]. The **augmented matrix** of a whole system is then a matrix with one row for each equation in the system.



__**II. Solving a System of Two Equations in Two Unknowns **__

We will use the following system of equations to demonstrate how to solve using matrices. **First step:** We need to create two matrices from the given system of equations. One of those matrices is referred to as the coefficient matrix. It is called the coefficient matrix because it is created by using the coefficients of the variables involved. So for this system, the coefficient matrix is: **Third Step:** Now we want to use these matrices to solve our system of equations. To do so, we must input the matrices in the graphing calculator. **Fourth Step:** To use these matrices to solve the system, we need to find the inverse of **[A]** and multiply that answer by **[B]**.
 * Second Step: ** We will create a constant matrix from the constants on the right side of the equal signs.
 * Find the MATRIX button on your graphing calculator
 * Go the the EDIT menu to input the matrices (If there is a matrix already in A or B, just simply type over it)
 * Select A and enter the dimensions of the matrices by typing.... 2 ENTER 2 ENTER (because it is 2 by 2)
 * Now enter the four numbers into the calculator. Press ENTER after each number and the calculator will automatically move you to the next number.
 * To input matrix B, go back to MATRIX....EDIT....MATRIX B and press ENTER
 * Type dimensions (like we did for matrix A)
 * Input Numbers and hit ENTER
 * Exit the MATRIX menu by hitting 2nd MODE
 * We are now ready to find the Inverse
 * Access the MATRIX menu
 * Make sure you are in the NAMES menu
 * Select **[A]** and press ENTER
 * Raise [A] by negative on to get its inverse.
 * Then, do the first three steps again except on the third one, choose **[B]**. It should look like this:




 * Now that the two matrices are next to each other, they are ready to be multiplied.

http://hmaricca.wikispaces.com/Period+6+-+Inverse+2x2
 * NOTE:** For further information on what the inverse of a 2x2 Matrix is, please go to this page which explains in detail what the inverse of a 2x2 matrix really means:


 * Fifth Step: **


 * Press ENTER to get the answers for x and y that solve our system.
 * You should get a matrix that looks like this:
 * The top digit of the matrix represents the x-value and the bottom represents the y-value. Therefore, x=1 and y=4.



__**III. More Difficult Problems**__ Solving a system of two equations using matrices in a word problem

The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make? Solving this system of equations using matrices.


 * Step 1: **

Let x = number of two point shots Let y = number of three point shots.

**Step 2:**

Create a system of equations

2x+3y = 80 x+y = 37

**Step 3:**

Enter those equations in as two matrices: the coefficients of the variables on the left sides of the equations being the digits in the first matrix (matrix A) and the constants on the rights side of the equations being the second matrix (matrix B). For linear systems, the number of rows in the matrices equal the number of equations there are and, for Matrix A, the number of columns equals the number of different variables there are in the equations (in this case, two). For Matrix B, for linear systems, there only needs to be one column for the constants. Overall, the matrices should look like this:

Matrix A:

Matrix B:

After you have typed in these matrices into your calculator, go into the Matrix,Name option and enter the two matrices you've just created. Then, raise Matrix A to the power of negative one to turn it into its inverse. Then, take Matrix B and the inverse of Matrix A and multiply them together. The resulting matrix should be a 2x1 matrix with the x-value as its top digits and y-value as its bottom.
 * Step 4: **

This is what the resulting Matrix should look like:

//Therefore, the x-value is 31 and the y-value is 6. Back to the basketball question, the answer is that the Lakers made 31 2-pointer shots and 6 3-pointer shots.//

**IV. More Practice Problems** // If you would like to test your knowledge on solving systems using matrices, here are a few examples to try! // 1. 2x+6y= 14 2x+33y=2

2. 4x-5y=33 -x+5y=-27

Answers: (1) x=-5, y=4 (2) x=2 , y=-5

http://www.sparknotes.com/math/algebra2/systemsofthreeequations/section2.rhtml http://people.hofstra.edu/Stefan_Waner/RealWorld/tutorialsf1/frames2_2.html http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml http://www.sparknotes.com/math/algebra2/matrices/summary.html
 * Websites:**