### Theory:

A solution means the value of an equation that satisfies the given equation.

In a two-variable equation, the solution also contains two values.

We use varies method to find the solution of a equation. Now we will see the two methods to find a solution of the equation.

We use varies method to find the solution of a equation. Now we will see the two methods to find a solution of the equation.

**Method I)**

Consider a equation with two variable: $x+2y=20$.

As there are two variables in the equation, here the solution is a pair of values, one for \(x\) and one for \(y\) which satisfy the given equation.

In the above equation by inception if we substitute \(x = 2\) and \(y = 9\) we get the result \(20\) as given in the equation.

$\begin{array}{l}x+2y=20\\ \\ (2)+2(9)=20\\ \\ 2+18=20\\ \\ 20=20\end{array}$

Here the solutions are \((0, 9)\) which are satisfy the $x+2y=20$ equation.

Here apart from the \((0, 9)\) the other values like \((10, 5)\) can also be a solution for the equation. We get the same answer by substituting as \(x = 10\) and \(y = 5\) in the given equation.

But on the other hand the values \((5, 10)\) is not the solution for the equation.

That is if we substitute \(x = 5\) and \(y = 10\) in the $x+2y=20$ equation, the result is not equal to \(20\) which is given in the equation.

$\begin{array}{l}x+2y=20\\ \\ (5)+2(10)=20\\ \\ 5+25=20\\ \\ 25\ne 20\end{array}$

Therefore \((5, 10)\) is cannot be the solution for the $x+2y=20$ equation.

**Method II)**

**Steps to find a solution of equation**:

**1.**To find solution of a equation first substitute \(x = 0\) and \(y = 0\)in the given equation.

**2.**Next perform an arithmetic operation to find the value of \(x\) and \(y\) which are the solution of given equation.

Consider a equation: $x+2y=20$.

**Step 1**: Substitute \(x = 0\) in the given equation.

That is,

$\begin{array}{l}x+2y=20\\ \\ (0)+2y=20\\ \\ 2y=20\\ \\ y=\frac{20}{2}\\ \\ y=\phantom{\rule{0.147em}{0ex}}10\end{array}$

Thus the value of (\(x, y)\) is \((0, 10)\) is the solution of $x+2y=20$.

Similarly substitute \(y = 0\) in the equation $x+2y=20$.

$\begin{array}{l}x+2y=20\\ \\ x+2(0)=20\\ \\ x=20\end{array}$

**Therefore**,

**the solution of the equation**$x+2y=20$

**is**\((0\),\(10)\)

**and**\((20\),\(0)\).

**Now we learned that**,

**in a two variable equation the equation contain at least two solutions which are satisfy the equation**.