**What is a straight line?**

**A straight line is a curve such that every point on the line segment joining any two points on it lies on it.**

Let ax + by + c = 0 be a first degree equation in x, y where a, b, c are constants. Let P (x_{1}, y_{1}) and Q(x_{2}, y_{2}) be any point on the curve represented by ax+ by + c = 0. Then,

ax_{1} + by_{1} + c = 0 and ax_{2} + by^{2} + c = 0

When we say that the first degree equation is x, i.e. ax + by + c = 0 represents a line, it means that all points (x, y) satisfying ax + by + c = 0lie along a line. Thus, a line is also defined as the locus of a point satisfying the condition ax + by + c = 0, where a, b, c are constants.

It should be noted that there are only two unknowns in the equation of a straight line because the equation of every straight line can be put in the form ax + by + c = 0, where a and b are two unknowns. It is important to note here that x and y are not unknowns. In fact, these are the coordinates of any point on the line and are known as current coordinates. Thus, **to determine a line we will need two coordinates to determine the two unknowns**.

**Slope of a Line**

**The trigonometrical tangent of the angle that a line makes with the positive direction of the x-axis in an anticlockwise sense is called the slope or the gradient of a line. **

The slope of a line is generally denoted by *m*. Thus m = tan*tan* θ

**Since a line parallel to the x-axis makes an angle of 0**^{o}** with the x-axis, therefore, its slope is tan 0**^{0}** = 0**

A line parallel to the y-axis, i.e. a line that is perpendicular to the x-axis makes an angle of 90^{o} with the x-axis, so its slope is tan $\frac{\pi}{2}$ = ∞. Also, the slope of a line equally inclined with axes is 1 or -1 as it makes an angle of 45^{o} or 135^{o} with the x-axis.

The angle of inclination of a line with the positive direction of the x-axis in an anticlockwise sense always lies between 0^{0} and 180^{0}.

The line of positive slope makes an acute angle with the parallel direction of the x-axis. The line of zero slope is parallel to the x-axis. The line of negative slope makes an obtuse angle with the parallel direction of the x-axis in an anticlockwise direction.

**Slope of a line in Terms of Coordinates of any Two Points on it**

What would be the slope of a line in terms of coordinates of any two points on it? Let us find out.

Let P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) be two points on a line making an angle θ with the positive direction of the x-axis. Draw Pl, QM perpendicular on x-axis and PN_QN on QM. Then,

PN = LM = OM – OL = x_{2} – x_{1} and QN = QM – NM = QM – PL = y_{2} – y_{1}

In △ PQN, we have,

tan θ = $\frac{QN}{PN} = \frac{y_2- y_1}{x_2- x_1}$

Thus, if (x_{1}, y_{1}) and (x_{2}, y_{2}) are coordinates of any two points on a line then its slope is given by

**m = **$\frac{y2- y1}{x2- x1} = \frac{Difference\: in\: ordinates}{Difference\: in\: adbscissae}$

Let us understand it through an example.

**Example**

Find the slope of a line that passes through the points (3, 2) and (-1, 5)

**Solution**

We know that the slope of a line passing through two points (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by

m = $\frac{y_2- y_1}{x_2- x_1}$

Here, y_{1} = 2, y_{2} = 6, x_{1} = 3, x_{2} = -1

Substituting these values in the given equation, we have

m = $\frac{5-2}{-1-3} = \frac{-3}{4}$

**Hence, the slope of a line that passes through the points (3, 2) and (-1, 5) is **$\frac{-3}{4}$

**How to use the slope calculator to find the slope of a line?**

It is quite simple to use the scientific notation calculator for performing operations involving scientific notations. The following steps are required to be followed for this purpose –

**Step 1 – **The first step is the know how of what the slope calculator looks like.** **The following is the snapshot of what the landing page would look like as soon as click on the “ Slope Calculator “ link –

**Step 2** – Above we can see clearly, that the slope calculator asks for some information to be entered. This information consists of the coordinates of the line that we wish to find the slope of. Recall that these coordinates are in the form of (x_{1}, y_{1}) and (x_{2}, y_{2}). Let us take two coordinates as an example. What would be the slope of a line if we have the coordinates ( 2 , 1 ) and ( -4 , – 5 ). Let us first calculate the same using the formula that we discussed above.

We will have, m = $\frac{y_2- y_1}{x_2- x_1}$

⇒ m = $\frac{-5-1}{-4-2} = \frac{-6}{-6}$ = 1

Now let us check the same using the slope calculator. As require, we will enter the values, ( 2 , 1 ) and ( -4 , – 5 ) in the box provided for (x_{1}, y_{1}) and (x_{2}, y_{2}) respectively. Below is the snapshot of how the values (x_{1}, y_{1}) and (x_{2}, y_{2}) would be entered –

**Step 3** – Now that we have entered all the required information, our last step is to perform the calculation. For this purpose, we just need to click on the “ calculate “ button. As soon as we will click on this button, we can see the result obtained on the right-hand side of the values that we had entered in the previous steps. Below is a snapshot of how the selection would look like when we will click on the “ calculate “ button –

We can see above that the slope calculator not only provides us with the values of the slope, y2- y1 and x2- x1, but it also provides a graphical representation of the straight line that would pass through these two points.

Repeated use of this slope calculator shall prove to be helpful in not only understanding the concept of the slope of a line but will also help understand how a slope is instrumental in defining a line on the graph.