Two equations representing the same line

Comparison of Equations

Before proceeding further, here is one important concept that you should remember.

Given a line’s equation, say 2x + 3y = 5, it doesn’t matter if we transform it by say, multiplying or dividing both sides by any constant (not zero), or adding the same quantity to both the sides.

That means 2x + 3y = 5, and 4x + 6y = 10 (obtained by multiplying both sides by 2) represent the same line.

All of these equations represent the same line as 2x + 3y = 5: 3y =5 – 2x, 20x + 30y = 50, 2x/5 + 3y/5 = 1.

Conversely, if two equations represent the same line, then one must be the other multiplied by some constant, or the ratios of the corresponding coefficients must be equal.

That is, if you’ve been given that 2x + 3y = 5 and Ax + By = C represent the same line then, \(\frac{2}{A}=\frac{3}{B}=\frac{5}{C}\)

It is a very common mistake to equate the coefficients directly, i.e. A=2, B=3 and C=5. This will lead to errors / wrong answers in many problems. I’ll come to this later.

Remember always to compare the coefficients and never to equate them !

In case the equations look like this, 3x – 4y + 4 = 0 and y = ax + b, then before comparing, make sure the equations are in the same form.

That is, either the second equation should be written as ax – y + b = 0 and then coefficients be compared, \(\frac{3}{a}=\frac{-4}{-1}=\frac{4}{b}\)

Or, the first one could be written as 4y = 3x + 4 (in the same form as y = ax + b), followed by comparing the coefficients.

To avoid errors or confusion, you can first transform both the equations to the form Ax + By + C = 0, and then compare.

Lesson Summary

  1. The equations ax + by + c = 0 and px + qy + r = 0 represent the same line if and only if a/p = b/q = c/r

This rule (i.e. comparison of coefficients) applies to any curve in general, as we’ll see in the subsequent chapters.

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