System of Linear Equations in Two Variables

A system of equations is when we have two or more equations are working together.

In linear equation in one variable, only one variable is present whose value is to be found out and only one equation is sufficient to find the value of the variable.

A linear equation in two variables has two variables in the equation. Solving this equation is not possible with one equation only. To solve the two variables, we need two equations in two variables.

Let us see what such pairs look like algebraically.

The general form for a pair of linear equations in two variables x and y is

a₁x + b₁y + c₁ = 0

and a₂x + b₂ y + c₂ = 0,

where a₁, b₁, c₁, a₂, b₂, c₂ are all real numbers and a₁² + b₁² ≠ 0, a₂² + b₂² ≠ 0

Some examples of pair of linear equations in two variables are:

2x + 3y – 7 = 0 and 9x – 2y + 8 = 0

5x = y and –7x + 2y + 3 = 0

x + y = 7 and 17 = y

There are three possibilities:

• No solution
• One solution
• Infinitely many solutions

"Independent" means that each equation gives new information. Otherwise, they are "Dependent".

Also called "Linear Independence" and "Linear Dependence"

Example:

x + y = 2

2x + 2y = 4

Those equations are "Dependent" because they are really the same equation, just multiplied by 2.

So, the second equation gave no new information.

Simultaneous Linear Equations are true at the same time. That means at a certain solution, all equations are true.

Or we can say, A pair of values of x and y satisfying each of the equations in the given system of two simultaneous equations in x and y is called a solution of the system.

Algebra vs Graphs

We prefer to use algebra where the system of linear equations have more than 2 variables and can’t be solved by a simple graph.

The solution of pairs of linear equations in two variables algebraically:

Solution by Substitution method:

Let the pair of equations be a₁x + b₁y + c₁ = 0 and a₂x + b₂ y + c₂ = 0.

1. From one of the equations, express one of the variables say y in terms of the other variable i.e., x.
2. Substitute the value of y, obtained in the above step, in other equation, getting an equation in x.
3. Solve the equation and get the value of x.
4. Substitute the value of x in expression for y obtained in the first step and get the value of y.

Solution by Elimination method, i.e., by equating the coefficients:

1. In the two given equations, make the coefficients of one of the variables numerically equal. To do so, multiply these coefficients by suitable constant.
2. Add or subtract the equations obtained in the above step according to as the terms having the same coefficients are of the opposite or of the same signs and get an equation in only one variable.
3. Solve the equation found and get the value of one of the variable.
4. Substitute the value of this variable in either of the two given equations and find the value of the other variable.

So, here you understood the concept of linear equation in two variables. Share your views in the comment box below.