Linear Equations in Two Variables

An equation which contains two variables and the exponents of each variable is one and has no term involving product of variables is called a linear equation in two variables.

For example, 2x + 3y = 4 and x - 2y + 2 = 3x + y + 6 are linear equations in two variables.

The general form of a linear equation in two variables is ax + by + c = 0 where  a, b and c are real numbers such that at at least one of a and b is non-zero.

A linear equation in two variables has more than one solution. For each value of y, we get a unique value of x. Thus, a linear equation in two variables will have infinitely many solutions.

Graph of Linear Equation in Two Variables

The graph of linear equation in two variables is a line and the coordinates of every point on the line satisfies the equation. If a point does not lie on the graph then its coordinates will not satisfy the equation.

From two given points, one and only one line can be drawn. Therefore, it is sufficient to take any two points, i.e., values of the variables x and y which satisfy the equation.

System of Linear Equations in Two Variables

The two equations taken together are called system of linear equations in two variables and the values of x and y which satisfy both equations simultaneously is called the solution.

There are different methods for solving such equation. These are graphical method and algebraic method.

Graphical Method

In this method, you have to draw the graphs of both linear equations on the same graph sheet. The graphs of the equations may be:

  1. Intersecting lines: In this case, the point of intersection will be common solution of both simultaneous equations. The x-coordinate will give the value of x and y-coordinate will given value of y. In this case system will have a unique solution.
  2. Coincident lines: In this case each point on the common line will give the solution. Hence, system of equations will have infinitely many solutions.
  3. Parallel lines: In this case, no point will be common to both equations. Hence, system of equations will have no solution.

Algebraic Method

Algebraic methods are:

  1. Substitution Method
  2. Elimination Method

These methods are useful in case the system of equations has a unique solution.

Substitution Method: In this method, we find the value of one of the variable from one equation and substitute it in the second equation. This way, the second equation will be reduced to linear equation in one variable.

Elimination Method: In this method, we eliminate one of the variable by multiplying both equations by suitable non-zero constant to make the coefficients of one of the variable numerically equal. Then, we add or subtract one equation to or from the other so that one variable gets eliminated and we get an equation in one variable.