You have seen that every linear equation in one variable has a unique solution.
In a linear equation involving two variables, As there are two variables in the equation, a solution means a pair of values, one for x and one for y which satisfy the given equation.
Let us consider the equation `2x + 3y = 12`. Here, `x = 3` and `y = 2` is a solution because when you substitute `x = 3` and `y = 2` in the equation above, you find that
`2x + 3y = (2 × 3) + (3 × 2) = 12`
This solution is written as an ordered pair `(3, 2)`, first writing the value for `x` and then the value for y. Similarly, (0, 4) is also a solution for the equation above.
On the other hand, `(1, 4)` is not a solution of `2x + 3y = 12`, because on putting `x = 1` and `y = 4` we get `2x + 3y = 14`, which is not `12`. Note that `(0, 4)` is a solution but not `(4, 0)`.
You have seen at least two solutions for 2x + 3y = 12, i.e., (3, 2) and (0, 4). Can you find any other solution? Do you agree that (6, 0) is another solution?
Verify the same. In fact, we can get many many solutions in the following way. Pick a value of your choice for x (say x = 2) in 2x + 3y = 12. Then the equation reduces to 4 + 3y = 12,
which is a linear equation in one variable. On solving this, you get ` y = 8/3 ` . So ` ( 2, 8/3) ` is another solution of 2x + 3y = 12. Similarly, choosing x = – 5, you find that the equation becomes `–10 + 3y = 12.`
This gives ` y = (22)/3 ` . So ` (-5, (22)/3 ) ` is another solution of `2x + 3y = 12`. So there is no end to different solutions of a linear equation in two variables. That is, a linear equation in two variables has infinitely many solutions.