Question

How do you write the equation in point slope form given $(2,5),\,(3,10)$ ?

Answer 1

In mathematics, the slope or gradient of a line defines both the direction and the steepness of the line. The letter “m” is often used to denote the slope. When a line lies in the plane containing x and y axes, the slope of a line is given by the change in the y-coordinate divided by the corresponding change in the x-coordinate between two distinct points of the line, that is, the slope of a line joining two points $({x_1},{y_1})$ and $({x_2},{y_2})$ is given by the formula $\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ .

We can solve the above question using this formula and the standard form of the line.

Complete step by step answer:
The equation of a line passing through two points $({x_1},{y_1})$ and $({x_2},{y_2})$ is given as –
$y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})$

In this question, we are given that $({x_1},{y_1}) = (2,5)$ and $({x_2},{y_2}) = (3,10)$ ; using these
values in the above formula, we get –
$y - 5 = \dfrac{{10 - 5}}{{3 - 2}}(x - 2) \\\ \Rightarrow y - 5 = \dfrac{5}{1}(x - 2) \\\ \Rightarrow y - 5 = 5x - 10 \\\ \Rightarrow y - 5x + 5 = 0 \\\$

Hence the equation in the point-slope form given $(2,5),\,(3,10)$ is $y = 5x - 5$

Note: The slope of a line is given as the tangent of the angle between the line and the x-axis. We know that the tangent function is equal to the ratio of the perpendicular and the base. In the graph, the perpendicular is equal to ${y_2} - {y_1}$ and the base is equal to ${x_2} - {x_1}$ so the slope is given as $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ . Now the standard equation of a line is given as $y = mx$ where m is the slope of the line, inserting the obtained value of m in this formula, we get – $y - {y_1} = (\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}})x - {x_1}$ . Using this approach, similar questions can be solved easily.