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:
"Independent" means that each equation gives new information. Otherwise, they are "Dependent".
Also called "Linear Independence" and "Linear Dependence"
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.
Solution by Elimination method, i.e., by equating the coefficients:
So, here you understood the concept of linear equation in two variables. Share your views in the comment box below.