# Identical Straight Lines

When the coefficients of two straight lines are proportional they are called identical straight lines.

Let us assume, the straight lines a$$_{1}$$ x + b$$_{1}$$ y + c$$_{1}$$ = 0 and a$$_{2}$$ x + b$$_{2}$$y + c$$_{2}$$  = 0 are identical then

$$\frac{a_{1}}{a_{2}}$$ = $$\frac{b_{1}}{b_{2}}$$ = $$\frac{c_{1}}{c_{2}}$$

To get the clear concept let us proof the above statement:

a$$_{1}$$x + b$$_{1}$$y + c$$_{1}$$ = 0 .…………………..(i)

a$$_{2}$$x + b$$_{2}$$y + c$$_{2}$$ = 0 .…………………..(ii)

Convert the straight line a$$_{1}$$x + b$$_{1}$$y + c$$_{1}$$ = 0 in slope-intercept form we get,

y = $$\frac{a_{1}}{b_{1}}$$x - $$\frac{c_{1}}{b_{1}}$$

Similarly, convert the straight line a$$_{2}$$x + b$$_{2}$$y + c$$_{2}$$ = 0 in slope-intercept form we get,

y = $$\frac{a_{2}}{b_{2}}$$x - $$\frac{c_{2}}{b_{2}}$$

If (i) and (ii) represent the equations of the same straight line then their slopes are equal.

i.e., - $$\frac{a_{1}}{b_{1}}$$ = - $$\frac{a_{2}}{b_{2}}$$

or, $$\frac{a_{1}}{a_{2}}$$ = $$\frac{b_{1}}{b_{2}}$$ .…………………..(iii)

Again, the y-intercepts of lines (i) and (ii) are also equal.

Therefore,  - $$\frac{c_{1}}{b_{1}}$$ = - $$\frac{c_{2}}{b_{2}}$$

or, $$\frac{b_{1}}{b_{2}}$$ = $$\frac{c_{1}}{c_{2}}$$ .…………………..(iv)

Therefore, from (iii) and (iv) it is clear that (i) and (ii) will represent the same straight line when

$$\frac{a_{1}}{a_{2}}$$ = $$\frac{b_{1}}{b_{2}}$$ = $$\frac{c_{1}}{c_{2}}$$.