Question

How do you write an equation of a line given point $(3,7)$ and slope $\dfrac{2}{7}$ ?

Answer 1

Hint: In this question, we need to determine an equation that passes through the given point with the slope. Firstly, we denote the given point as $({x_1},{y_1})$. We make use of the formula of the equation of a line in point slope form given by $y - {y_1} = m(x - {x_1})$. We know the point and slope, so we substitute it in the given formula and simplify it. Then we obtain the desired equation of a straight line.

Complete step by step answer:
The point from which the line passes is given by $(3,7)$ and the slope of the line is given as $\dfrac{2}{7}$.
We are asked to write an equation of a line that passes through the given point with the given slope.
Now let us first understand the meaning of the term slope.
Slope is nothing but the ratio of y coordinate and x coordinate. Therefore, the slope of the line is given by $\dfrac{y}{x}$.
We denote the given point as $({x_1},{y_1}) = (3,7)$
And the slope as $m = \dfrac{2}{7}$
Note that for any line the slope is constant.
To find the equation of a straight line, we make use of the formula of equation of a line in point slope form which is given by,
$y - {y_1} = m(x - {x_1})$
where $m$ is the slope of the line and $({x_1},{y_1})$ is the point from which the line passes.
Here we have $({x_1},{y_1}) = (3,7)$ and $m = \dfrac{2}{7}$.
Substituting this values in the formula, we get,
$ \Rightarrow y - 7 = \dfrac{2}{7}(x - 3)$
Multiplying the terms in the R.H.S. and simplifying we get,
$ \Rightarrow y - 7 = \dfrac{2}{7}x - \dfrac{2}{7}(3)$
$ \Rightarrow y - 7 = \dfrac{2}{7}x - \dfrac{6}{7}$
Adding 7 to both sides of the equation, we get
$ \Rightarrow y - 7 + 7 = \dfrac{2}{7}x + 7 - \dfrac{6}{7}$
$ \Rightarrow y + 0 = \dfrac{2}{7}x + \dfrac{{49 - 6}}{7}$
Simplifying the above expression, we get,
$ \Rightarrow y = \dfrac{2}{7}x + \dfrac{{43}}{7}$

Therefore, the equation of the line that passes through the point $(3,7)$ with the slope $\dfrac{2}{7}$ is given by, $y = \dfrac{2}{7}x + \dfrac{{43}}{7}$.

Note: Note that the equation of a straight line can be written in various forms. But for a given problem, we must know which formula we need to use, so that the simplification will be easier. The form used above is called the point slope form.
The point slope form is given by, $y - {y_1} = m(x - {x_1})$
where $m$ is the slope of the line and $({x_1},{y_1})$ is the point from which the line passes.
We can solve the above problem by using slope intercept form, which is given by,
$y = mx + c$
where $m$ is the slope and $c$ is the intercept.