Suppose a line goes through the points (1, 7) and (–2, –8). Write the equation of the line. Determine the intercepts of the line and?

Suppose (x, y) is a point in the straight line.
(y - 7)/(x - 1) = (-8 - 7)/(-2 - 1)
(y - 7)/(x - 1) = 5
y - 7 = 5(x - 1)
y - 7 = 5x - 5
Equation of the straight line : 5x - y + 2 = 0

When x = 0 :
5(0) - y + 2 = 0
y = 2
Then, the y-intercept = 2

When y = 0 :
5x - (0) + 2 = 0
x = -2/5
Then, the x-intercept = -2/5

Finally, sketch a straight line which passes through (1, 7), (-2, -8), (0, 2) and (-2/5, 0).

The line through (1, 7) and (-2,-8) is given by (1--2)(y--8) = (7--8)(x--2) [yields 3(y+8)=15(x+2)]
which can be manipulated to (7--8)x - (1--2)y = 7×-2 - 1×-8 [yields 15x-3y=-6] in Standard Form.
Hence the x intercept is (7×-2 - 1×-8)/(7--8) [yields -2/5]
and the y intercept is (1×-8 - 7×-2)/(1--2) [yields 2]

Graph: https://www.desmos.com/calculator/k3oeuproez

Rise over run gives you slope. That's all you need to know to figure it out. That and y = mx + b, m is the slope and b is the y axis intercept because that's when x is zero. To find the x intercept set y to zero and solve for x. How can we play with numbers like this? Because a line is infinite and we're only looking to explain it from two points and then figure out where that laser beam strikes the vertical and horizontal lines in its way.

An equation of the line can be written as
.. (-8-7)(x -1) -(-2-1)(y -7) = 0
.. -15x +15 +3y -21 = 0 . . . eliminate parentheses
.. 5x -y +2 = 0 . . . divide by -3
.. x/(-2/5) +y/(2) = 1 . . . divide by -2, add 1. This is intercept form

The standard form equation for the line is 5x -y = -2.
The x-intercept is -2/5.
The y-intercept is 2.

A graph can be seen at the source link.
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Click on the equation or the line to highlight the intercepts. Click on those to see their coordinates.