Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given y = 5x - 7 and -4 + y = -1 ?

Answer 1

The one solution, the point where the lines represented by the two equations cross, is the point `(2,3)`.

Explanation:

These equations are for straight lines. If they cross there will be one solution - the point where they cross. If they are parallel there will be no solutions. If they are the same line there will be many (infinite) solutions.

First step is to get them both into standard form. Standard form for a line is:

`y=mx+b` where `m` is the slope (gradient) and `b` is the y-intercept.

The first line given is already in this form:

`y = 5x-7`

For the second one, we rearrange to make y the subject:

`y = 3` (add 4 to both sides)

This is a horizontal line at `y=3`, so it has a slope of `0`.

Since the other line has a slope of `5`, they (a) are not parallel (or they'd have the same slope) and (b) are not the same line.

That means there will be one solution. We can already tell its `y` value will be 3, since one of the lines has all its points at `y=3`. To find the `x` value of the intersection, substitute `y=3` into the first equation and solve for `x`:

`y = 5x-7`
`3 = 5x-7`
`5x=10`
`x=2`

So the one solution, the point where the lines cross, is the point `(2,3)`.

You could also solve or check this by drawing a graph of the two lines and noting where they cross.