Linear equation in one variable: Linear equation of the form ax + b = 0, where a, b are real numbers such that a≠ 0. For example, 3x + 5 = 0. The letters used in an equation are called variables or unknowns. In the equation 3x + 5 = 0;
x is called the variable of the equation.
Linear equation in two variable:
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. Let us consider the following equations:
(i) 3x + y = 7
(ii) 4x – y = 3
(iii) √2x - √3y = 9
(iv) y = 2x + 5
Note : If in the equation ax+ by = c; we have a = 0 and b≠ 0 or b = 0 and a≠ 0 .Then the equation reduces to by = c or ax = c. In both case we get an equation in one variable. So, it is generally assumed that both a and b are non-zero as given in the above definition.
Remember: A linear equation in two variables has infinitely many solutions.
• Graph of a linear equation in two variables:
The graph of every linear equation in two variables is a striaght line.
Algorithm: To draw the graph of a linear equation in one variable we may follow the following steps:
Step I: Obtain the linear equation.
Step II : If the equation is of the form ax = b; a ≠ 0 then plot the point (b/a,0) and one more point (b/a, β) where β is any real number on the graph paper.
If the equation is of the form ay= b, a ≠ 0 then plot the point (0,b/a) and (α,b/a), where α is any real number on the graph paper.
Step III : Join the points plotted in step II to obtain the required line.
•The point (a, 0) where the straight line cross x-axis is called its x-intercept point and ‘a’ is called x-intercept.
•The point (0, b) where the straight line crosses y-axis is called its y-intercept point and ‘b’ is called y-intercept.
• x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.
•The graph x = a is a straight line parallel to the y-axis.
•The graph y = a is a striaght 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. More over every solution of the linear equation is a point on the graph of the linear equation.