In this lesson, we’ll talk about what it means for two (different) equations to represent the same line.
For example, if the equations ax + by = c and 2x – 3y = 5 represent the same line, then what can be the values of a, b, and c? Tempted to say a = 2, b = -3, and c = 5? Don’t rush yet!
Suppose we have an equation: 2x – 3y = 5. Then, it doesn’t matter if we transform it by multiplying or dividing both sides by any non-zero number, or adding the same number to both sides.
For example, 2x – 3y = 5 and 4x – 6y = 10 (obtained by multiplying both sides by 2) represent the same line.
Infact, all these equations represent the same line as 2x – 3y = 5:
-3y = 5 – 2x (subtracting 2x from both sides)
20x – 30y = 50 (multiplying both sides by 10)
2x/5 – 3y/5 = 1 (dividing both sides by 5)
2x – 3y + 4 = 9 (adding 4 to both sides)
Conversely, if two equations represent the same line, then one must be the same as the other multiplied by some non-zero constant. In other words, the ratios of the corresponding coefficients must be equal.
For example, if 2x – 3y = 5 and ax + by = c represent the same line, then:
2/a = -3/b = 5/c
To find particular values of a, b, and c, we can fix any one of these, and then solve for the other two.
For example, if c = 5, then we’ll get a = 2 and b = -3. If c = -10, then a = -4 and b = 6, and so on. So, we’ll have infinitely many combinations of a, b, and c, each of which will give us a line, which represents the same line as 2x – 3y = 5.
We can also equate each of these fractions to k. That is:
2/a = -3/b = 5/c = k
Now, we can choose any non-zero value of k, and then solve for a, b, and c.
Suppose the equations (representing the same line) look like a little ‘different’, like this:
3x – 4y + 4 = 0
y = ax + b
Then, before comparing, we’ll make sure the equations are in the same ‘form’.
That is, either the second equation should be written as ax – y + b = 0 (which looks similar to 3x – 4y + 4 = 0) and then coefficients can be compared:
3/a = (-4)/(-1) = 4/b
Or, the first one can be written as 4y = 3x + 4 (in the same form as y = ax + b), followed by comparing the coefficients:
4/1 = 3/a = 4/b
We’ll get the same equations in both cases. To avoid confusion, you can first transform both the equations to the form ax + by + c = 0, and then compare.
Lesson Summary
- 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. The next lesson will talk about intersection and concurrency of lines.