Two rails are represented by the …

We have,
{tex}x + 2y - 4=0{/tex}

Putting {tex}y = 0{/tex}, we get
{tex}x + 0 - 4 = 0{/tex}

{tex} \Rightarrow {/tex} {tex}x = 4{/tex}
Putting x = 0, we get
{tex}0 + 2y - 4 = 0{/tex}

{tex} \Rightarrow {/tex} {tex}y = 2{/tex}
Thus, two solutions of equation {tex}x + 2y - 4 = 0{/tex} are:

We have, 
{tex}2x + 4y - 12 = 0{/tex}
Putting {tex}x = 0{/tex}, we get
{tex}0 + 4y - 12 = 0{/tex}

{tex} \Rightarrow {/tex} {tex}y = 3{/tex}
Putting {tex}y = 0{/tex}, we get
{tex}2x + 0(12) = 0{/tex}

{tex} \Rightarrow {/tex} x = 6
Thus, two solutions of equation {tex}2x + 4y - 12 = 0{/tex} are:

Now, we plot the points A (4, 0) and B (0, 2) and draw a line passing through these two points to get the graph of the line represented by the equations (i).
We also plot the points P (0, 3) and Q (6, 0) and draw a line passing through these two points to get the graph of the line represented by the equation (ii).
We observe that the lines are parallel and they do not intersect any where.

REMARK The graphical representation of the above pair of linear equations provides us a pair of parallel lines.
Let us write the pair of linear equations,
{tex}x + 2y - 4 = 0{/tex}
{tex}2x + 4y -12 = 0{/tex}
as {tex}a_1x + b_1y + c_1=0{/tex}
{tex}a_2x + b_2y +c_2 =0{/tex}
where {tex}a_1=1, b_1= 2, c_1 = -4{/tex},

{tex}a_2 = 2, b_2 = 4\ and \ c_2 = -12{/tex}
{tex}\frac { a _ { 1 } } { a _ { 2 } } = \frac { 1 } { 2 } , \frac { b _ { 1 } } { b _ { 2 } } = \frac { 2 } { 4 } = \frac { 1 } { 2 } \text { and } \frac { c _ { 1 } } { c _ { 2 } } = \frac { - 4 } { - 12 } = \frac { 1 } { 3 }{/tex}
{tex}\therefore \quad \frac { a _ { 1 } } { a _ { 2 } } = \frac { b _ { 1 } } { b _ { 2 } } \neq \frac { c _ { 1 } } { c _ { 2 } }{/tex}
will represent parallel lines, if
{tex}\frac { a _ { 1 } } { a _ { 2 } } = \frac { b _ { 1 } } { b _ { 2 } } \neq \frac { c _ { 1 } } { c _ { 2 } }{/tex}
The converse is also true for any pair of linear equations.
It follows from the above examples that the pair of linear equations
{tex}a_1x + b_1y + c_1 = 0{/tex}
{tex}a_2x + b_2y +c_2=0{/tex}
will represent:

1. intersecting lines, if {tex}\frac { a _ { 1 } } { a _ { 2 } } \neq \frac { b _ { 1 } } { b _ { 2 } }{/tex}
2. coincident lines, if {tex}\frac { a _ { 1 } } { a _ { 2 } } = \frac { b _ { 1 } } { b _ { 2 } } = \frac { c _ { 1 } } { c _ { 2 } }{/tex}
3. parallel lines, if {tex}\frac { a _ { 1 } } { a _ { 2 } } = \frac { b _ { 1 } } { b _ { 2 } } \neq \frac { c _ { 1 } } { c _ { 2 } }{/tex}