Solving system of linear equations in two or three variables

Linear equations are equations of the first order. This equation is defined for the lines in the coordinate system. The equation that has a homogeneous variable of degree 1 (i.e. only one variable), is known as a linear equation in one variable. A linear equation can have more than one variable, If it has two variables, then it is called a linear equation in two variables, and so on. For example- linear equations are 2x – 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, 3x – y + z = 3.

It is a part of the derivatives and functions of an equation. A linear equation consists of derivatives of multiple variables. When both the result and variable are partial, then it is also said as a linear partial differential equation. 

Condition for Unique Solution to Linear Equations

A system of two linear equations ax + by + c = 0 and dx + ey + g = 0 will have a unique solution, if and only if the two lines represented by the equations intersect at a point. The equation of two lines is neither parallel nor coincident. Moreover, the slopes of the two lines should be different.

Expressing both of the equations in the standardized y = mx + c format, where ‘m’ is the slope of the line and ‘c’ is the y-intercept.

ax + by + c = 0 can be written as y =  −ax/b−c/a
similarly, dx + ey + g = 0 can be written as y =  −dx/e−g/e
The slope of the first line is  −a/b and that of the second line is  −d/e
For a unique solution, the slopes of the lines should be different.
∴  −a/b≠−d/e
Or a/d≠b/e

Why do we use matrices to solve a system of linear equations?

In the linear equation of one variable let’s say 5x = 2x + 3. We can easily get  the value of  

x = 1. 

When we have a pair of linear equations consisting of two variables, we use linear equations concept i.e elimination method, substitution method, cross multiplication method etc to find out the value of the unknown variables.

 Similarly, if we have a  number of linear equations consisting of number of variables, then the process to find the value of the unknown variables becomes complex. 

So, to solve such equations, use of inverse matrix is comparatively easier. This article tries to clear the concept of applying matrix inverse methods for solving a system of equations. The concepts are explained with the help of examples for an easy understanding.

By understanding this concept, students will understand how to use the given inverse to solve the system of equations. 

Inverse Matrix to solve the system in Linear Equations

Here is a detailed explanation of the way to use an inverse matrix to solve the linear system. 

For using the inverse matrix, we require equations-

• e1x+f1y+g1z=d1
• e2x+f2y+g2z=d2
• e3x+f3y+g3z=d3

Here d1, d2, d3 are non-zero values.

Let A be  e1 e1 e3 f1 f2 f3 g1 g2 g3             

B be x y z                                 

And C be d1 d2 d3  

When taking the inverse matrix to solve a linear system, we require a coefficient matrix, B as a coefficient of a linear system.

This makes the following equation as 

e1 e1 e3 f1 f2 f3 g1 g2 g3  x x y z =d1 d2 d3

By considering the above array, one can conclude the equation of a system to be 

Ax = B

This further gives two cases that can be explained when using the inverse to solve a system of equations. This method of determining the inverse requires two additional matrices. The variables and the constants are represented by Matrix A and B respectively. 

Let’s take A as a coefficient matrix, X as a variable matrix, and B as a constant matrix to solve a system of linear equations with an inverse matrix to solve the system AX=B.

Case 1: Here A is a matrix in the non-singular form

As we know the fact that the value of a matrix’s determinant can’t be zero. This proves that the inverse of a matrix subsists in an equation.

Following the above-said statement, we can write Ax = B

To find the value, we multiply both the sides by A-1

This gives the value A-1(Ax) that is equal to A -1B

Using associative property, it gives (A-1A) x which is equal to A -1B

So, the value of x is A-1B.

Now we can find a solution due to the inverse nature of the matrix in a linear equation. 

Case 2:  Here A is a singular matrix.

It is known to us that a singular matrix has the value of zero in the case of determinants. Therefore, the value of adjacent (A)B can be found.

As adjacent (A)B is not equal to 0, which makes the solution void, thus making the equation of the system inconsistent.

Let’s take a case where adj(A)B is equal to zero, which gives the value of a linear equation to be consistent or inconsistent. This is due to multiple solutions or a lack of one.

Conclusion

In this article, we have learned about the meaning of linear equations, the use of matrices in linear equations, inverse of a matrix to solve the linear equation with a unique solution. We have also learned to check whether the equation has a unique solution or not. The matrix method is suitable to find out the solution to the system of the equations. Here, all the variables should be written in the proper order at the appropriate sides, write the variables, their coefficients, and constants.