# How do you find the slope given A(2, -4) and B(4, 7)?

May 8, 2017

Slope = $\frac{11}{2}$

#### Explanation:

The slope, gradient or steepness of a line is defined as:

The $\text{the vertical change"/"the horizontal change}$ which is written as

$\frac{\Delta y}{\Delta x} = \text{change in y"/"change in x}$

If you have any two points on a line you can find the slope of the line between them by using the formula:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} \text{ }$ Either point can be $\left({x}_{1} , {y}_{1}\right)$

We have $A \left(2 , - 4\right) \mathmr{and} B \left(4 , 7\right)$

$m = \frac{7 - \left(- 4\right)}{4 - 2} = \frac{7 + 4}{2} = \frac{11}{2}$

The slope is left in this form. This slope represents a very steep line which has a vertical increase of 11 for a horizontal change of only 2.

The line with this slope through the given points is shown below:
graph{y = 11/2x -15 [-44.56, 115.54, -32.3, 47.7]}

May 8, 2017

$+ \frac{11}{2}$

#### Explanation:

$\textcolor{b l u e}{\text{Preamble}}$

The slop (gradient) is the amount up or down for a given amount of along reading left to right on the x-axis.

If the slop is like going up a hill then it is positive. On the other hand if it is like going down a hill then it is negative.
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$\textcolor{b l u e}{\text{Answering the question}}$

Looking at the x's; the left most point is A as 2 is less than 4.
So we read from point A to point B

Let point 1 be ${P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(2 , - 4\right) \to A$
Let point 2 be ${P}_{2} \to \left({x}_{2} , {y}_{2}\right) = \left(4 , 7\right) \to B$

The gradient $m \to \left(\text{change in y")/("change in x}\right) \to \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Some people write this as $\frac{\Delta y}{\Delta x} = m$ where

$\Delta y$ means change in $y$ and $\Delta x$ means change in $x$

$\textcolor{b r o w n}{\text{So we have:}}$

$\text{gradient } \to m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{7 - \left(- 4\right)}{4 - 2} = + \frac{11}{2}$

As positive it means the 'slope' is upwards reading left to right. 