Question

The equation of a line is $y = mx + 1$ . How do you find the value of the gradient $m$ given that $P(3,7)$ lies on the line?

Answer 1

Hint: The first thing to note here is that we can find a second point that lies on this line by making $x = 0$, i.e.., by staring at the worth of the y-intercept. Then find the slope with the help of two given points on the line.

Complete step-by-step solution:
The problem tells you that the equation of a given line in slope-intercept form is ,
$y = mx + 1$
The first thing to note here is that we can find a second point that lies on this line by making $x = 0$, i.e.., by staring at the worth of the y-intercept.
$
   \Rightarrow y = m.(0) + 1 \\
   \Rightarrow y = 1 \\
 $
This means that the point $(0,1)$ lies on the given line.
And the point $P(3,7)$ also lies on the given line. (given)
Now, the slope of the line , $m$ , are often calculated by gazing at the ratio between the change in $y$, and also the change in $x$.
Slope of a line can also be found if two points on the line are given . let the two points on the line be $({x_1},{y_1}),({x_2},{y_2})$ respectively.
Then the slope is given by , $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ .
Therefore ,
$
   \Rightarrow m = \dfrac{{7 - 1}}{{3 - 0}} \\
   \Rightarrow m = \dfrac{6}{3} \\
   \Rightarrow m = 2 \\
 $
This means that the slope of the plane is $m = 2$ .

Additional Information:
Slope of a line can also be found if two points on the line are given . let the two points on the line be $({x_1},{y_1}),({x_2},{y_2})$ respectively.
Then the slope is given by , $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ .
Slope is also defined as the ratio of change in $y$ over the change in $x$ between any two points.

Note: A straight-line equation is always linear and represented as $y = mx + c$ where $m$is the slope of the line and $c$ is the y-intercept and $\dfrac{{ - c}}{m}$ is the x-intercept .
y-intercept is calculated by substituting $x = 0$.
Similarly, x-intercept is calculated by substituting $y = 0$ .