# Graphically Solving a Pair of Linear Equations

We have seen that for any linear equation, the graph we get is a straight line. This means that the coordinates of every point on this line will satisfy the linear equation. Now, suppose that we take a pair of linear equations:

\[\begin{array}{l}{a_1}x + {b_1}y + {c_1} = 0\\{a_2}x + {b_2}y + {c_2} = 0\end{array}\].

Suppose that we draw the lines corresponding to both these equations on common axes, and obtain the following graph:

What is the geometrical interpretation of this? The two lines intersect at the point *A*. Clearly, since *a* lies on both the lines, the coordinates of *A*, that is, \(\left( {h,\;k} \right)\), will satisfy both the linear equations. Thus,

\[\begin{array}{l}{a_1}h + {b_1}k + {c_1} = 0\\{a_2}h + {b_2}k + {c_2} = 0\end{array}\]

Technically speaking, we will say that the coordinates of *A* form the **solution to the pair of linear equations** given by (1). Sometimes, we might also omit the phrase “*coordinates of*” and simply state: *A* is a solution to the pair of linear equations given by (1). It should be understood that by *A*, we mean the coordinates of *A*.

As an example, consider the following pair of linear equations:

\[\begin{array}{l} - x + 2y - 3 = 0\\ 3x + 4y - 11 = 0\end{array}\]

We draw the corresponding lines on the same axes:

The point of intersection is \(A\left( {1,\,\,2} \right)\), which means that \(x = 1,\;\;y = 2\) is a solution to the pair of linear equations given by (2). In fact, it is *the only* solution to the pair, as two non-parallel lines cannot intersect in more than one point.