Chapter 11, Problem 62RE

To determine

To find: The four ways of solving a system of linear equations containing three variables, the preferred method and reason behind it.

Answer to Problem 62RE

The four methods, preferred method and the reason behind preference is discussed below..

Explanation of Solution

Given information:

The system of linear equation in a three variable.

Calculation:

The first method of solving a linear equation in three variables is Method of Substitution.

In this method the value of one variable is solved in terms of other two variables and similarly the other values are calculated and then these values are substituted in the equations to get the value of variables.

The second method of solving a system of linear equations in three variables is by using the method of elimination.

In this method the variables are eliminated by making the common coefficient followed by adding or subtracting the equations from each other.

The third method is Method of determinant. In this method the determinant is calculated and if the value of determinant is non zero then $D_x$ , $D_y$ and $D_z$ is calculated.

Then below formulas are used.’

  $\begin{array}{l}x=\frac{D_x}{D}\\ y=\frac{D_y}{D}\\ z=\frac{D_z}{D}\end{array}$

The fourth method to solve the system of linear equation with three variables is use of inverse matrix.

The matrix is obtained by coefficients of variables in the three equations and then the inverse matrix is obtained and the column matrix is obtained.

The solution is obtained from the elements of column matrix which is obtained by product of inverse matrix and column matrix.

The inverse matrix method is preferred while solving a system of linear equations in three variables because it tells whether the solution is a set of finite numbers or infinite solutions and it is simpler method.

Therefore, the four methods, preferred method and the reason behind preference is discussed above.