Question

How do I create a linear equation with a given solution like (3,0)?

Answer 1

There are infinitely many such linear equations. Choose any `b` or `m` and plug it into `b=–3m` to get the other. Your equation will be `y=mx+b` with these values of `m` and `b.`

Explanation:

You want `(3,0)` to be a solution to your linear equation. That means the line must pass through the point `(3,0).` In other words, we need to find a slope and `y`-intercept so that the equation `y=mx+b` works when `(x,y)=(3,0).`

`y=mx+b`
`0=m(3)+b`

Solving this for `b`, we get

`b = –3m`

Ending with an equation like this that links `b` and `m` means that there are an infinite number of slope-intercept pairs that produce a line that passes through `(3,0).` Indeed, this should make sense; we can imagine pivoting a pencil around that fixed point.

To create a single line as an example, just choose any random value for either `b` or `m`, plug it into `b=–3m`, and solve for the other.

Example: The equation of a line that passes through `(3,0)` with slope `m=2` is found like this:

`b=–3m`
`color(white)b=–3(2)`
`color(white)b=–6`

The equation for such a line is `y=2x-6`.