Question

What is an equation of the line that goes through point (8, −9) and whose slope is undefined?

Answer 1

The equation of a line is `x=8`.

Explanation:

General equation of a straight line on a coordinate plane is
`ax+by+c=0`
where `a`, `b` and `c` - some real constants and variable `x` and `y` are coordinates.

In order for this line to go through point `(8,-9)`, values `x=8`, `y=-9` should satisfy this equation:
`a*8+b*(-9)+c=0`

Since another condition is that a slope is undefined, it means that the line should be parallel to Y-axis, that is any other point with the same X-coordinate (`x=8`) and any Y-coordinate (say, `y=0` or `y=1`) also lies on this line and, therefore, should satisfy the equation.

For instance, let's use point `(8,0)`, which produces:
`a*8+b*0+c=0`
And for point `(8,1)`:
`a*8+b*1+c=0`
And any other point of a type `(8,y)` should produce
`a*8+b*y+c=0`

This necessitates `b=0` and the only condition on coefficients `a` and `c` is:
`a*8+c=0`

Any real numbers that satisfy the above conditions, for instance `a=1, c=-8` or `a=2, c=-16` etc., are good to be coefficients to construct an equation of a line that goes through point `(8,-9)`.

Examples are:

1. `a=1, b=0, c=-8`
   Equation: `x-8=0`

2. `a=2, b=0, c=16`
   Equation `2x-16=0`

Obviously, all equations are equivalent to each other and can be reduced to a form `x=8`.