● As the following are examples of linear equations in two variables:
`2x + 3y = 5`
`x – 2y – 3 = 0`
and `x – 0.y = 2`, i.e., `x = 2`
● You also know that 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.` (We often denote the condition a and b are not both zero by `a^2 + b^2 ≠ 0`).
`=>` For example, let us substitute `x = 1` and `y = 1` in the left hand side (LHS) of the equation `2x + 3y = 5`. Then
LHS `= 2(1) + 3(1) = 2 + 3 = 5`, which is equal to the right hand side (RHS) of the equation.
● Therefore, x = 1 and y = 1 is a solution of the equation `2x + 3y = 5`.
Now let us substitute x = 1 and y = 7 in the equation `2x + 3y = 5`. Then,
`LHS = 2(1) + 3(7) = 2 + 21 = 23` which is not equal to the RHS.
● Therefore, `x = 1` and `y = 7` is not a solution of the equation.
● Geometrically, It means that the point `(1, 1)` lies on the line representing the equation `2x + 3y = 5`, and the point (1, 7) does not lie on it. So, every solution of the equation is a point on the line representing it.
● In fact, this is true for any linear equation, that is, each solution `(x, y)` of a linear equation in two variables, `ax + by + c = 0`, corresponds to a point on the line representing the equation, and vice versa.
Now, consider Equations (1) and (2) given above. These equations, taken together, represent the information we have about Akhila at the fair.
● These two linear equations are in the same two variables `x` and `y.` Equations like these are called a pair of linear equations in two variables.
`=>` Let us see what such pairs look like algebraically.
● The general form for a pair of linear equations in two variables x and y is
`a_(1) x + b_(1) y + c_1 = 0`
and `a_(2) x + b_(2) y + c_2 = 0`,
where `a_1, b_1, c_1, a_2, b_2, c_2` are all real numbers and `a_(1)^2 + b_(1)^2 ≠ 0 , a_(2)^2 + b_(2)^2 ≠ 0` .
● Some examples of pair of linear equations in two variables are:
`2x + 3y – 7 = 0` and `9x – 2y + 8 = 0`
`5x = y` and `–7x + 2y + 3 = 0`
`x + y = 7` and `17 = y`
● You have also studied in Class IX that given two lines in a plane, only one of the following three possibilities can happen:
(i) The two lines will intersect at one point.
(ii) The two lines will not intersect, i.e., they are parallel.
(iii) The two lines will be coincident.
We show all these possibilities in Fig. 3.1:
In Fig. 3.1 (a), they intersect.
In Fig. 3.1 (b), they are parallel.
In Fig. 3.1 (c), they are coincident.
`=>` Both ways of representing a pair of linear equations go hand-in-hand—the algebraic and the geometric ways. Let us consider some examples.