# Solution of a Linear Equation

We observed that every linear equation in one variable has a unique solution. The solution of a linear equation in two variables is a pair of values (x, y) which satisfy the given equation. Consider an equationÂ 2x + 5y = 17. Here, x = 1 and y = 3 is a solution as when we substitute x = 1 and y = 3 in the equation, we get

2x + 5y = 2Â´1 + 5Â´ 3 = 2 + 15 = 17

Thus the solution is (1, 3). Similarly (6, 1) is also a solution for the above equation. Try the pair (0, 3). L.H.S = 2Â´0 + 5Â´3 = 0 + 15 = 15, which is not 17. Hence (0, 3) is not a solution to the above equation. But this doesnâ€™t mean that there are no more solutions. We can get many solutions in the following way:Â

Â

Let x = 3 and 2 Â´ 3 + 5y = 17 i.e., 5y = 17 â€“ 6 Ãž 5y = 11 Ãž y = .

.

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.

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**Example : **

**1)** Find four solutions of the equation x + 3y = 7

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Let us choose x = 1 then 3y =6 and hence y =2

therefore x = 1 and y =2__ __is a solution of the equation x + 3y = 7

If we take x =4 then 3y =3 or y =1

therefore x = 4 and y =1 is a solution of the equation x + 3y = 7

If we assume__ __x =0 then y = ,is a solution of the equation x + 3y = 7

Â If we assume y = 0 then x = 7, is a solution of the equation x + 3y = 7

(1, 2), (4, 1), (0, ), (0 ,7) are the four solutions of the infinitely many solutions.

**2)** Find two solutions of the equation 5x + 3y = 8

Let us assume x=1 then 3y =8 - 5 =3y=3Ãž .

Â x=1 and y = 1 is a solution of the equation 5x + 3y = 8

Let us assume x = 0 then y =

x = 0 and y = is a solution of the equation 5x + 3y = 8.

Hence (1, 1) and (0, ) are the two solutions of the infinitely many solutions.

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