System of Linear Equations Using Determinants

The equation of a straight line is a linear equation. It consists of variables of power 1. Linear equations are also called first-degree equations. For example, 2x + y = 1 is a linear equation. Here x and y are variables of power 1. As far as the JEE exam is concerned, the linear equation is an important topic. A set having two or more linear equations is called a system of linear equations.

There are different methods to solve linear equations. In this article, we will learn how to solve a system of linear equations using determinants. This method is also called Cramer’s rule. Some of the important methods are given below.

• Graphing method
• Substitution method
• Linear combination method
• Matrix method
• Using determinants

Solving System of Linear Equations using Determinants

Consider two equations

a₁x + b₁y = c₁

a₂x + b₂y = c₂

The solution to the above equations is given by

x = D₁/D

y = D₂/D

D is the determinant of the variables.

Steps to Solve the System of Equations

1. Calculate the determinant D, using the coefficients of the variables.

2. Calculate the determinant D₁, using the constants in place of the x coefficients.

3. Calculate the determinant D₂, using the constants in place of the y coefficients.

4. Find x and y using equations x = D₁/D and y = D₂/D.

5. Write the solution as an ordered pair.

Solving System of Three Equations with Three Variables.

Consider three equations

a₁x + b₁y + c₁z = k₁

a₂x + b₂y + c₂z = k₂

a₃x + b₃y + c₃z = k₃

The solution to the above equations is given by

x = D₁/D

y = D₂/D

z = D₃/D

While solving a system of three equations with three variables with Cramer’s Rule, we do what we did for a system of two equations. Now we have to solve for three variables to get the solution. Let us have a look at an example of solving equations with two variables using the determinant method.

Example:

Solve 3x + 4y = 5

2x – y = 7

Solution:

Given 3x + 4y = 5

2x – y = 7

Here

= 3×-1-2×4

= -3-8

= -11

= 5×-1-7×4

= -5-28

= -33

= 3×7-2×5

= 21 – 10

= 11

x = D₁/D

= -33/ -11

= 3

y = D₂/D

= 11/-11

= -1

Hence the solution is (3,-1).

Conditions for Infinite and No Solutions

1. If D = 0, and D₁ = D₂ = D₃ = 0, then the system of equations may or may not be consistent.

2. If the value of x, y, and z satisfy the third equation then the system is said to be consistent and will have infinite solutions.

3. If the values of x, y, z do not satisfy the third equation, the system is said to be inconsistent and will have no solution.

4. If D₁ = D₂ = D₃ = 0, then the system of linear equations is called homogeneous linear equations, which will have at least one solution i.e. (0, 0, 0). This is known as a trivial solution for homogeneous linear equations.

5. If the system of homogeneous linear equations has non-zero solutions and D = 0, then the given system has infinite solutions. Then we can o solve the equations using the matrix inversion method.

Linear Equations In Algebra

Algebra is a vast subject having many applications. Linear algebra is used in fractal geometry, differential equations, difference equations, relativity, archaeology, demography, etc. Students can expect questions fromlinear equations in algebra for the JEE exam. The standard form of a linear equation in two variables is ax + by = c, where a and b are real numbers. The variables are x and y.

While solving the equations in one variable, we follow the rules like addition rule, subtraction rule, multiplication rule, and division rule. According to the type of solutions, the system of equations can be consistent or inconsistent.