Question

How do you write the equation of a straight line using point slope form if \[m = 6\] and point \[\left( {4, - 5} \right)\]?

Answer 1

Hint: The slope of line is given to us along with a point from which the line is passing through. We have to use the point slope form of the equation of line to find its equation. The point slope form of line is $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$, where $m$ is the slope and $\left( {{x_1},{y_1}} \right)$ is the point. Put the given values in this and find the equation.

Complete step-by-step answer:
According to the question, we have to find the equation of a straight line using point slope form with the slope and the point from which it is passing is given.
We know that the point slope form of the equation of line is given as:
$ \Rightarrow \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right){\text{ }}.....{\text{(1)}}$
Here, $m$ is the slope of line and $\left( {{x_1},{y_1}} \right)$ is the point through which it is passing. These are already given in the question for the required line.
So we have \[m = 6\] and the point is \[\left( {4, - 5} \right)\]. Thus $\left( {{x_1},{y_1}} \right)$ is equivalent to \[\left( {4, - 5} \right)\]. Putting these values in equation (1), we have:
$
   \Rightarrow \left( {y - \left( { - 5} \right)} \right) = 6\left( {x - 4} \right) \\
   \Rightarrow y + 5 = 6x - 24 \\
   \Rightarrow 6x - y = 29 \\
 $

Therefore the equation of line is $6x - y = 29$.

Note:
There are other methods also to find the equation of a straight line. Some of them are shown below:
(1) If x and y intercepts of a line are known to us, we can easily determine the equation of line. Let $a$ and $b$ are the x and y intercepts respectively of the line, then the equation of line is:
$ \Rightarrow \dfrac{x}{a} + \dfrac{y}{b} = 1$
(2) And if only y intercept is known along with the slope of line then also its equation can be easily determined. If $m$ is the slope and $c$ is the y intercept of the line, then the equation of line is:
$ \Rightarrow y = mx + c$