Point-slope form is just one of many ways of describing a line which states that, given the slope of a line and a point on that line, you have enough information to completely describe that line.
A line is defined by some constant rate of change - its slope. If m is the slope of a line, then we can define m as
[tex]m = \frac{y-y_1}{x-x_1} [/tex]
Where [tex](x_1,y_1)[/tex] is some fixed point and [tex](x,y)[/tex] is some other point on the line that we can vary. Here, we know that m = 4, and we're given a fixed point of [tex](-1,3)[/tex] to work with, so we have
[tex]4= \frac{y-3}{x-(-1)}= \frac{y-3}{x+1} [/tex]
Multiplying both sides by [tex]x+1[/tex], we get the equation in point-slope form:
[tex]4(x+1)=y-3\\ y-3=4(x+1)[/tex]
To put the equation in slope-intercept form, we need to find the point where the line hits the y-axis. To do this, we can simply set x = 0:
[tex]y-3=4(0+1)\\ y-3=4\\ y=7[/tex]
So, the line hits the y-axis at [tex](0,7)[/tex]. Given our slope is 4, and given the general slope-intercept form of a line,
[tex]y=mx+b[/tex]
Where m is our slope and b is our y-intercept, the slope-intercept form for our line would be
[tex]y=4x+7[/tex]
