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The concept of slope is extremely important so it is not sufficient to know how you calculate it using

.

In simple terms, the slope specifies the units by which the co-ordinate changes and the direction in which it changes with 1 unit increase in co-ordinate. If the slope, , is positive, the co-ordinate changes in the same direction as the co-ordinate.

If is negative, the co-ordinate changes in the opposite direction.

For example, if the slope of a line is 2, it means this:

If the co-ordinate increases by 1 unit, the co-ordinate will increase by 2 units. So if the point (3, 5) lies on a line with slope 2, the point (4, 7) will also lie on it. Here, when the co-ordinate increases from 3 to 4, the co-ordinate will increase from 5 to 7 (increase of 2 units). Similarly, the point (2, 3) will also lie on the same line. If the co-ordinate decreases by 1 unit (from 3 to 2), the co-ordinate will decrease by 2 units (from 5 to 3). Since the slope is positive, the direction of change of co-ordinate will be the same as the direction of change of co-ordinate.

Similarly, if the slope of a line is -2 and the point (3, 5) lies on it, when the co-ordinate increases by 1 unit, the co-ordinate DECREASES by 2 units. So the point (4, 3) will also lie on the line. Similarly, if the co-ordinate decreases by 1 unit, the co-ordinate will increase by 2 units. So the point (2, 7) will also lie on the line.

This understanding of the slope concept can be very helpful as we will see in this question.

Line and line have slopes -2 and respectively. If line and line intersect at (6,8), what is the distance between the -intercept of line and the -intercept of line ?

(A) 5

(B) 10

(C)

(D) 15

(E)

**Solution:**

Traditionally, one would solve this question like this:

**Method 1:** Slope of line formula

Equation of a line with slope m and constant is given as .

The equations of lines and would be

and

As both these lines pass through (6,8), substitute and to get the values of and .

Thus equations of the 2 lines become

and

intercept of a line is given by the point where . So intercept of line is given by

So line intersects the axis at the point (10, 0)

intercept of a lien is given by the point where . So intercept of line is given by

So line intersects the axis at the point (0, 5)

Distance between these two points is given by

Answer **(C)**.

**Method 2:** Using Slope Concept

Now notice how we will solve this question using the concept we discussed above:

For line :

Slope = -2 means that for every 1 unit increase in co-ordinate, co-ordinate decreases by 2. Line has slope -2 and passes through (6, 8). It’s intercept will have i.e. a decrease of 8 so will increase by 4 to give . So intercept is at (10, 0).

Line has slope and passes through (6, 8). It’s intercept will have i.e. a decrease of 6 in co-ordinate. This means will decrease by of that i.e. by 3 and will become . So intercept is at point (0, 5)

Distance between the two points can be found using Pythagorean theorem as

or Distance between two points formula

Answer **(C)**.

Using the slope concept makes solving this question much less tedious and we save a lot of time. That is the advantage of holistic approaches over the more traditional approaches.