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?

Jan 9, 2017

$m = 2$

Explanation:

The problem tells you that the equation of a given line in slope-intercept form is

$y = m \cdot x + 1$

The first thing to notice here is that you can find a second point that lies on this line by making $x = 0$, i.e. by looking at the value of the $y$-intercept.

As you know, the value of $y$ that you get for $x = 0$ corresponds to the $y$-intercept. In this case, the $y$-intercept is equal to $1$, since

$y = m \cdot 0 + 1$

$y = 1$

This means that the point $\left(0 , 1\right)$ lies on the given line. Now, the slope of the line, $m$, can be calculated by looking at the ratio between the change in $y$, $\Delta y$, and the change in $x$, $\Delta x$

$m = \frac{\Delta y}{\Delta x}$

Using $\left(0 , 1\right)$ and $\left(3 , 7\right)$ as the two points, you get that $x$ goes from $0$ to $3$ and $y$ goes from $1$ to $7$, which means that you have

$\left\{\begin{matrix}\Delta y = 7 - 1 = 6 \\ \Delta x = 3 - 0 = 3\end{matrix}\right.$

This means that the slope of the line is equal to

$m = \frac{6}{3} = 2$

The equation of the line in slope-intercept form will be

$y = 2 \cdot x + 1$

graph{2x + 1 [-1.073, 4.402, -0.985, 1.753]}