An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables. A linear equation in two variables has infinitely many solutions.
The graph of every linear equation in two variables is a straight line. x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis. The graph of x = a is a straight line parallel to the y-axis. The graph of y = a is a straight line parallel to the x-axis.
An equation of the type y = mx represents a line passing through the origin.
Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph of the linear equation.
Class 10
An equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables x and y. Solution of such an equation is a pair of values, one for x and the other for y, which makes the two sides of the equation equal.
Two linear equations in the same two variables are called a pair of linear equations in two variables. The most general form of a pair of linear equations is
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
A pair of linear equations in two variables can be represented, and solved, by the:
- Graphical method
- Algebraic method
Graphical Method: The graph of a pair of linear equations in two variables is represented by two lines.
- If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.
- If the lines coincide, then there are infinitely many solutions - each point on the line being a solution. In this case, the pair of equations is dependent (consistent).
- If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.
Algebraic Methods: Following methods are used for finding the solutions of a pair of linear equations:
- Substitution Method
- Elimination Method
- Cross-multiplication Method
For a pair of linear equations, following situations can arise:
- a1/a2 ≠ b1/b2 : In this case, the pair of linear equations is consistent.
- a1/a2 = b1/b2 ≠ c1/c2 : In this case, the pair of linear equations is inconsistent.
- a1/a2 = b1/b2 = c1/c2 : In this case, the pair of linear equations is dependent and consistent.
There are several situations which can be mathematically represented by two equations that are not linear to start with. But we alter them so that they are reduced to a pair of linear equations.