Cramer's+3x3

Jordan Bloom Sean Hynekamp Philip Yoo

For background information on Gabriel Cramer himself, check out Period 8's Cramer's Rule 3x3 page.

What is Cramer's Rule Used For?
Cramer's Rule for a 3x3 System is a convenient way to solve for one of the variables in a system of equations without having to solve for the other variables. For example, when given a system of equations such as:

**2**//**x**// **+** //**y**// **+** //**z**// **= 3** //**x**// **–** //**y**// **–** //**z**// **= 0** //**x**// **+ 2**//**y**// **+** //**z**// **= 0**

With the use of matrices, you will be able to find the value for any of these variables, in this case x, y or z.

How do you use Cramer's Rule?
Steps: > //** 2 x**// **+** //** 1 y**// **+** //** 1 z**// **= 3 ** > //** 1 x**// **–** //** 1 y**// **–** //** 1 z**// **= 0 ** > //** 1 x**// **+ //2// **//**y**// **+** //** 1 z**// **= 0 ** > Of course, if a variable is not shown in an equation, its coefficient is zero. > > But also, keep in mind the constants (the right side of the equation) and keep them in a 3x1 matrix fashion: > > > > > > > > > **x = Dx / D = 3 / 3 = 1** > **y = Dy / D = -6 / 3 = -2** > **z = Dz / D = 9 / 3 = 3**
 * 1) First, take a look at the coefficients of the variable as well as the values of the constants in the above system of equations.
 * 1) Create a 3x3 matrix of the coefficients respective to their positions in the system of equations like so:
 * 1) Now, replace the first column of the matrix with the constants' "matrix" for the X matrix, the second column for the Y matrix, and the third column for the Z matrix as shown below. As you can see, the coefficients of x are replaced in the X matrix, the coefficients of y replaced in the Y matrix, and the coefficients of z replaced in the Z matrix.
 * 1) Next, find the determinants of each matrix (the coefficients, X, Y, and Z). You can learn how to find the determinant in Period 6's very colorful Determinant page.
 * 1) To find the solutions and values of x, y, and z of the systems of equations divide the determinant of the variable's respective matrix by that of the coefficient matrix.
 * 1) TA-DA! You have found the solution of the system of equations {x = 1, y = -2, z = 3}.


 * TIPS AND EXTRA TIDBITS**
 * Remember to be wary of the dreaded signs! Put parentheses around negative numbers so you don't make a sign mistake.
 * If the determinant of the coefficient matrix is 0, the solution is either "no solution" (it is inconsistent) or "infinitely many solutions" (it is dependent). You may have to manually determine if the solution is either "no solution" or "infinitely many solutions" by looking at the coefficients of the system of equations. Remember the coefficients of a system of equations that has "infinitely many solutions" are proportional to each other (i.e. x + y + z = 1, 2x + 2y + 2z = 2, 3x + 3y + 3z = 3).
 * You can always check your answers by plugging the values of the variables back into the equations.

In the work shown, ignore the left hand column and the top row. The real matrix is just 3x3 table of number values. The extra column and row are there for reference. Work and answers are here: https://docs.google.com/document/d/1WDW9HaStiVzek1qFHGV_duCtm42nEBmZZA_Bl77Avo0/edit > **4x + 2y + 3z = 10** > **6x + y + 3z = 10** > > ** x + y + 2z = 7 ** > ** x + 3y + 4z = 14 ** > The JAR file was sent to Ricca. Graphics classes and code credited to Brian Reiff. Actual Cramer's Rule code by Philip Yoo. Download here
 * Extra Practice Problems and Answers with Work**
 * 1) **2x + y + z = 4**
 * 1) ** y + z = 5 **
 * Cramer's Rule Java Program**

Sources:
http://www.purplemath.com/modules/cramers.htm