Q. 62 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

The four methods of solving system of three linear equations with three variables is 

1. Method of substitution

2. Method of elimination

3. Determinant method

4. Use of inverse matrix

And I prefer Inverse matrix since it is more easy and easy to interpret if the equation is consistent or not.

Step 1. Given

A system of three linear equations with three variables.

Step 2. Substitution method.

any one of the three equations should be converted into equations with two variables and solve the equation for one variable.

Then substitute the known variables to find the unknown variables.

Step 3. Elimination method

Among the three variables in the system of equations, any one of the variable should be canceled by using any of the basic operations and find the other variables.

The substitute the known variable in any of the equation to find the unknown variable.

Step 4. Determinant method.

Determinant method consists of the coefficients of the variables of the equation and the solution exist only if the determinant is non-zero.

The solutions of the equation are given by the expression below:

$$\begin{gathered} x=\frac{D_x}{D}; \\ y=\frac{D_y}{D} \\ z=\frac{D_z}{D} \end{gathered}$$x

Step 5. Use of Inverse matrix

Find the inverse of the given matrix

The coefficients of variables is placed in the rows and columns and we must operate either row or column until we get the echelon form.

Then by solving the matrix, we will arrive at the solution.

Step 6. The method I prefer

Among four methods, the first two methods are sometimes difficult according to the given system of equations.

And the method, Use of Inverse Matrix is the most preferable one.

Since it is easy to reduce and less time consuming and the constraints can describe us whether the given system of equations has solutions or inconsistent.