The line through point $(m, -9)$ and $(7, m)$ has slope m. The y-intercept of this line, is?

We know that the slope $m$ of a line passing through two points

$(x_1,y_1)$ and

$(x_2,y_2)$

is $m=\dfrac{y_2-y_1}{x_2-x_1}$.

Let $(m,-9)=(x_1,y_1)$ and

$(7,m)=(x_2,y_2)$, then the slope of the line can be determined as:

$m=\dfrac { y_{ 2 }-y_{ 1 } }{ x_{ 2 }-x_{ 1 } } \\ \Rightarrow m=\dfrac { m-(-9) }{ 7-m } \\ \Rightarrow m=\dfrac { m+9 }{ 7-m } \\ \Rightarrow m(7-m)=m+9$

$\Rightarrow { -m }^{ 2 }+7m=m+9\\ \Rightarrow { m }^{ 2 }-7m+m+9=0\\ \Rightarrow { m }^{ 2 }-6m+9=0\\ \Rightarrow { (m-3) }^{ 2 }=0\\ \Rightarrow m-3=0\\ \Rightarrow m=3$

Therefore, the points are $(3,-9)$ and $(7,3)$.

We also know that the equation of the line passing through a point

$(x_1,y_1)$

is $(y-y_1)=m(x-x_1)$ where $m$ is the slope of the line.

Thus, the equation of the line passing through the point

$(7,3)$ is as follows:

$(y-y_{ 1 })=m(x-x_{ 1 })\\ \Rightarrow (y-3)=3(x-7)\\ \Rightarrow y-3=3x-21\\ \Rightarrow 3x-21-y+3=0\\ \Rightarrow 3x-y-18=0\\ \Rightarrow 3x-y=18$

Now, put $x=0$ in the above equation to get the y-intercept as:

$(3\times 0)-y=18\\ \Rightarrow 0-y=18\\ \Rightarrow y=-18$

Hence, the y-intercept of the line is $y=-18$.