Linear Equations - How To Predict The Nature Of Solution!

In a system of two linear equations, there are three possibilities about the solution of the system.

In this presentation, I am going to explore all of three possibilities one by one.
1. The equation have one unique solution: The system of two equations can have one unique solution. First of all the students need to know the meaning of one unique solution for the two linear equations.

One unique solution means, if we draw both the linear equations on the graph, we get two straight lines which might intersect at a point on the coordinate frame.
The point of intersection is called the solution of the equations, which gives the value of both the variables. Both the linear equations can have one solution if their slopes are different.
For example; consider we have the following system of linear equations: 3x + y = 2 -2x + y = -9 To find the slope we have to solve both the equations for "y" as shown below: First equation is changed to slope and y-intercept form as y = - 3x + 2 The coefficient of "x" which is "- 3" is the slope of line and constant term "2" is called the y-intercept. Similarly, second equation can be changed to slope and y-intercept form as shown below: y = 2x - 9 Slope = 2 and y-intercept = - 9 for this line.
Now, slope of first line is "- 3" and that of second line is "2". Therefore, both the lines have different slopes and hence have one unique solution.

In other word there is one unique number value for variable "x" and another number value for "y" or in other words, if these lines are drawn on the grid, both lines will intersect at a point.

2. The equations have no solution: There is another possibility, that the equations can't be solved and we can't find the values of variables, which is called the equations have no solution.
The process to detect this possibility is same as in case one.

Slopes and y-intercepts of both the lines are obtained and if both lines have the same slopes but different y-intercepts, then they have no solution. No solution means, if both lines are drawn on the grid they will be parallel to each other and never intersect with each other.
3.
The linear equation got infinite many solutions: The third possibility is that both the equations got infinite many solutions. This is the case when both the equations got same slopes and same y-intercepts. If the lines are drawn on the coordinate grid, they will overlap each other and each point is a solution for the system.

Hence, a system of two linear equations in two variables can have above three possibilities about their solutions.
The nature of the solution can be predicted without solving the equation by finding slopes and y-intercepts.