Question

How do you determine the slope of a line? Is there more than one way to determine the slope? Why or Why not? How do you find the intercepts of a line? Explain using an example.

Answer 1

The slope of a line is defined as the rate of change of dependent variable with respect to the independent variable. There are different approaches to get the slope of a given line but the value remains same in all cases. This is because the rate can be calculated using any two points for a straight line.

Let's say you have a line : aX+bY+c=0.

Then the first and elemental way of getting the slope is by identifying two points which lie on this line say(x₁, y₁) and (x₂, y₂). Then slope is defined as -

Example - Let's say we have 2x + 3y - 6 = 0. Put x = 0 to get y = 2 and put y = 0 to get x =3. So, the two points lying on the line are (0,2) and (3,0).

So, the slope would be -

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Rearrange the given equation in form of Y = mX + C. Then 'm' will be slope of your line.

So, if you are given the line equation as aX + bY + c = 0, then rearrange it as-

So, the slope would be -

Considering the same example of 2x + 3y - 6 =0, we can rearrange it to get -

Then the slope is m = -2/3.

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The third approach involves calculus. The rate of change is defined as -

So, you can differentiate the function implicitly as -

So, you can apply this to the example we have been considering so far i.e. 2x + 3y - 6 = 0.

Differentiate it to get -

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Note that all the values are same. So, there might be different approaches to get the slope but the value remains same.

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Intercept of a line is the value of the dependent variable 'Y' when the independent variable takes the value x = 0.

So, if you have the line aX + bY + c = 0, then the intercept would be -

For example if you have the line 2x + 3y - 6 = 0, then the intercept would be -

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