Period+8+Cramer's+Rule+for+a+2x2+System

James, Billy, Nick

=**Summary of Cramer's Rule for a 2x2 System**=
 * Cramer's Rule for a 2x2 System provides a convenient way to solve a system of equations containing two variables without having to use the traditional methods of substitution or elimination.
 * It can be summed up like this:

>> dependent.
 * Basically what this says is that x is equal to the determinant of y's coefficients and the constants (b1, b2, c1, c2) divided by the determinant of the coefficients of x and y (a1, a2, b1, b2). Similarly, y is equal to the determinant of x's coefficients and the constants (a1, a2, c1, c2) divided by the determinant of the coefficients of x and y (a1, a2, b1, b2).
 * If the determinant in the denominator is 0, then one of two things is true:
 * 1. If all the determinants in the above formulas are zero (both numerators and the denominator), then the system is
 * 2. If the denominator is zero and there is at least one numerator that is not zero, then the system is inconsistent.

=**Steps for Using Cramer's rule in a 2 x 2 System**= (3 8) (4 5) (25 8) (22 5) (3 25) (4 22)
 * This will be a walk through for the system 3x + 8y = 25 and 4x + 5y = 22
 * The first step to solving this is to find the determinant of the coefficients in the system. Every time we need to find a determinant, we must create a 2x2 square matrix with certain elements. In this case, the matrix consists of the coefficients of x and y.
 * Following the steps to find the determinant, we multiply 3 and 5 making 15 and subtract the product of 4 and 8, which is 32, which leaves us with -17.
 * Therefore, the determinant of this is -17. We'll call this value D.
 * Now we have to find another determinant. It's the same process as finding D, except now the matrix consists of the constants of the equations and the coefficients of y.
 * (25 x 5) = 125
 * (22 x 8) = 176
 * The determinant of this is 125 - 176 = -51. We'll call this value Dx.
 * To find the value of x, all we need to do is divide Dx by D
 * x = -51 / -17 = 3
 * Now we have to find the final determinant of this system. Follow the same steps except now fill the matrix with the coefficients of x and the constants of the equations.
 * (3 x 22) = 66
 * (4 x 25) = 100
 * The determinant of this is is 66 - 100 = -34. We'll call this value Dy.
 * To find y, we divide Dy by D
 * y = -34 / -17 = 2
 * Therefore, by using Cramer's Rule, we can easily find the solution to this system to be (3,2).
 * Cramer's Rule can definitely be a useful way to solve 2x2 systems especially when traditional methods such as substitution or elimination are not convenient. However, they come in handy the most when solving systems with more variables such as 3x3 systems.

=**Program**=

Here is a program that uses Cramer's rule to solve a 2x2 system: (enter values for the coefficients and constants in both equations and hit solve)

=**Video**=

This youtube video provides a simple explanation of Cramer's Rule if you still need more help: media type="youtube" key="7bsRfjVQ6z8" height="315" width="560" = =

Bibliography: marc.mtsac.edu/worksheet/general_topics/7cramer_rule.pdf @http://www.coolmath.com/algebra/14-determinants-cramers-rule/01-determinants-cramers-rule-2x2-03.htm