Chapter 11, Problem 110RE

To determine

Which method do you prefer?

Answer to Problem 110RE

  $\boxed{\text{Inverse Matrix}}$

Explanation of Solution

Given information:

Describe four ways of solving a system of three linear equations containing three variables. Which method do you prefer? Why?

Calculation:

The system of linear equations in three variables can be solved using four methods.

The first method to solve an equation in three variables is by Method of Substitution.

We obtain a relation in terms of any of the two variables and substitute in the rest of the two equations. Now we obtain two linear equations in two variables. The system of equations in two variables can further be solved using Method of Substitution for a linear equation in two variables.

The second method to solve an equation in three variables is by Method of Elimination.

In this method, we tend to eliminate one of the variables and obtain the two equations in two variables which can be solved further by Method of Elimination for two variables. The values of the variables obtained can be substituted in either of the equations to obtain the value of the third variable.

The above two methods discussed are cumbersome and it gets difficult to handle such equations. The third method is based on determinants which are quite an elegant method to determine the solutions to the linear equations in three variables.

We obtain a determinant which is composed of the coefficients of the three variables. Firstly we determine the value of this determinant and we proceed further only if the value is non-zero. Then we obtain the determinants $D_x,D_{y}and{D}_z$

Column obtained from the constants in the three given equations. The solutions of the equations are given by the expression given below.

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

The fourth method is form the use of inverse matrix of the matrix composed of the coefficients of the variables in the three given equations. Then we obtain the inverse of the obtained matrix. The column matrix composed of the constants from the three given equations is also determined in order to obtain the solutions.

The solutions to the three given equations are given by the elements of the column matrix obtained from the product of the inverse Matrix and the Column matrix obtained previously.

The best method to solve the equations is from the use of inverse Matrix because when the inverse of the matrix obtained from the elements of the coefficients of the variables in the three given equations is not feasible, the solutions are not possible which means the equations are inconsistent.

Hence, the best way to solve a system of equations by using $\boxed{\text{Inverse Matrix}}$ .