Solution of a Linear Equation

The solution of a linear equation is defined as the set of all possible values to the variables that satisfies the given linear equation.

Types of Solutions for Linear Equations

There are 3 possible types of solutions to the set of linear equations and are mentioned below.

• Unique Solution
• No Solution
• Infinitely Many Solutions

Unique Solution

The linear equation in one variable always has a unique solution. The unique solution of a linear equation represents that there exists only one point, which on substitution, L.H.S becomes equal to R.H.S. In the case of simultaneous linear equations in two variables, the solution should be an ordered pair $(x,y)$. In this case, the ordered pair will satisfy the set of equations.

Example: $3x+2=11$

$3x=11-2=9$

$3x=9$

$x=3$

Thus, the unique solution of the given linear equation is $x=3$

No Solution

If the graphs of the linear equations are parallel, then the system of linear equations has no solution. In this case, there exists no point such that no lines intersect each other.

Example: Find the Solutions to the equations $-2x+y=9$ and$-4x+2y=5$?

Solution:

The equations $-2x+y=9$ and $-4x+2y=5$ have no solution.

The line equations $-2x+y=9$ and $-4x+2y=5$ are parallel to each other, and hence, they do not have solutions.

Infinitely Many Solutions

A linear equation in two variables has infinitely many solutions. For the system of linear equations, there exists a solution set of infinite points for which the L.H.S of an equation becomes R.H.S. The graph for the system of linear equations with infinitely many solutions is a graph of straight lines that overlaps each other.

Example: Find four different solutions of the equation $x+2y=6$

$x$	$y$	$x+2y$
2	2	6
0	3	6
6	0	6
4	1	6

Four different solutions are $(2,2),(0,3),(6,0),(4,1)$