and a , b , c denote the coefficents of equations given in the general form in Section . . Table . Sl Pair of lines Compare the Graphical Algebraic No.
ratios representation interpretation . x – y = ≠ Intersecting Exactly one x + y – = lines solution (unique) . x + y – = Coincident Infinitely x + y – = lines many solutions . x + y – = ≠ Parallel lines No solution x + y – = From the table above, you can observe that if the lines represented by the equation a x + b y + c = a x + b y + c = are (i) intersecting, then ≠ ⋅ (ii) coincident, then ⋅ (iii) parallel, then ≠ ⋅ In fact, the converse is also true for any pair of lines.
You can verify them by considering some more examples by yourself. Let us now consider some more examples to illustrate it. Example : Check graphically whether the pair of equations x + y = x – y = is consistent. If so, solve them graphically.
Solution : Let us draw the graphs of the Equations ( ) and ( ). For this, we find two solutions of each of the equations, which are given in Table . Fig. .
Table . y = y = x − – – Plot the points A( , ), B( , ), P( , – ) and Q( , – ) on graph paper, and join the points to form the lines AB and PQ as shown in Fig. . .
We observe that there is a point B ( , ) common to both the lines AB and PQ. So, the solution of the pair of linear equations is x = and y = , i.e., the given pair of equations is