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Cramer’s rule

20 Oct 2015Homework Help Online

Cramer’s rule is applicable on the linear equations. So, for better understanding of the chapter, it’s important to understand the fundamentals of it.

What are linear and non linear equations?

Linear equations: It is a first order equation involving two variables; its graph is a straight line in the coordinate system. Whereas, the equation which doesn’t form a straight line is called a nonlinear equation.

Example of a linear equation:

2x –y +z = 0

Example of a nonlinear equation:

3x2 + 2y –z = 0

Now, that you are clear with the difference between linear as well as nonlinear equations it’s easy to understand the application of crammer’s rule. Let’s study what crammer’s rule is.

In linear equations, Cramer's Rule is an open method to solve a system of linear equations with numerous equations when the equation has a unique solution. Cramer’s rule checks you from solving the entire system of equations by solving only one variable. Let's use the following system of equations:

3x+y+z=3

x+y+z=0

Benefits of Cramer’s rule:

Cramer’s rule saves your energy and time both in one go. Once familiar with the Cramer’s rule,  you need not to solve the whole system to get the desired value. Be it math’s exercises or physics tests Cramer’s rule is all rounder. It adjusts everywhere to prove its unique nature.

Simple steps to learn Cramer’s rule:

Ø   You just pick the variable to be solved.

Ø   Replace that variable's column of values in the coefficient determinant with the answer-column's values,

Ø  Calculate that determinant

Ø  Divide it by the coefficient determinant.

Follow four simple steps and learn the Cramer’s rule to avail its benefits at every step.

The Determinant:

Ø  There is another method to solve systems of equations, involving a quantity called determinants. Every m×m matrix has a unique determinant. The determinant is a single number.

Ø   To find the determinant of a 2×2 matrix, multiply the numbers on the downward diagonal and subtract the product of the numbers on the upward diagonal.

Examples:

x + 2y + 3z = 1

-x + 2z = 2

-2y + z = -2

Determinants and Crammer’s for 2 * 2 systems

x- 2y =4

5x+7y =8

For the following system of equations, find the value of z.

2x +   y +   z = 1

x –   y + 4z = 0

x + 2y – 2z = 3

To solve only for z, I first find the coefficient determinant.

Then I form Dz by replacing the third column of values with the answer column:

Then I form the quotient and simplify:

Answer: Z = 2

Wasn’t it an easy chapter to understand? Now, it’s your turn. All you ne4ed to do is solve every equation applying the Cramer’s rule and walk parallel with the crowd of your class.

Useful links to study more about the Cramer’s rule

  1. Use the link to learn more about Cramer’s rule
  2. Use Mathportal for the complete steps of Cramer’s formula. It can help you solve the problems online and get the solutions in detail.
  3. Study all the formulas of Cramer’s rule.
  4. Use the link to study the chapter in detail with examples.
  5. See Wikipedia for all the history as well as the complete chapter of Cramer’s rule.

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