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Find the slope of the line passing through the points A(2,3) and B(4,7).

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Hint: The slope is also known as gradient which describes the direction and the steepness of the line. The slope of a line in the plane containing the X and Y axis is generally represented by the letter m, and is defined as the change in the Y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. Mathematically, it is expressed as following,

\[\begin{align}

&\text{we have two given points }({{x}_{1}},{{y}_{1}})\text{ and }({{x}_{2}},{{y}_{2}})\text{ then slope ''m'' is given by,} \\
&\text{m= }\tan \theta =\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \\
\end{align}\]
Here, \[\theta \] is the angle made by the line joining the two given points with the X axis in an anticlockwise direction.


Complete step-by-step answer:
In the given question we have \[{{x}_{1}}=2,{{y}_{1}}=3\text{ and }{{x}_{2}}=4,{{y}_{2}}=7.\]
\[\begin{align}
 & \text{we have two given points }({{x}_{1}},{{y}_{1}})\text{ and }({{x}_{2}},{{y}_{2}})\text{ then slope ''m'' is given by,} \\
 & \text{m= }\tan \theta =\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \\
\end{align}\]
So, the slope “m” is as follow;
\[\begin{align}
 & \Rightarrow m=\dfrac{7-3}{4-2} \\
 & \Rightarrow m=\dfrac{4}{2}=2 \\
\end{align}\]
Hence, the slope will be equal to 2.
Therefore, the correct answer of the above question is 2.

NOTE: Remember the formula to calculate the slope when two points are given which is also shown below:
We have two given points $({{x}_{1}},{{y}_{1}})\text{ and }({{x}_{2}},{{y}_{2}})$ then the slope “m” is given by,
\[m=\tan \theta =\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
The concept of slope is very important in mathematics as well as in engineering as it applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope of the line is related to its angle of incline \[\theta \] by the tangent function which I have mentioned in the solution.
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