What is the solution to `{:(–x+2y=4),(5x-3y=1):}`?

Each of these equations represents a line in 2D space. As with any pair of lines, they may cross, they may be parallel, or they may be the same line. Solving a pair of equations simultaneously means finding the `(x, y)` point where the lines cross (if it exists).

We start by assuming there is a point `(x, y)` that works for both equations

`"-"x+2y=4`
`5x-3y=1`

If this is true, then we can rearrange each equation and combine the two equations together to help us narrow in on the coordinates of the `(x,y)` point.

For example, if `"-"x+2y=4`, then we have

`color(blue)(x=2y-4)`

by solving for `x`. But, if this is the same `x` that works for the other equation, we can substitute this expression for `x` into the other equation like this:

`"      "5color(blue)x"      "-3y=1`
`5(color(blue)(2y-4))-3y=1`

and we end up with an equation with just `y`. Thus, we can solve for `y`:

`10y-20-3y=1`
`color(white)(10y-20-)7y=21`
`color(white)(10y-20-7)color(red)(y=3)`

So, this is the `y`-coordinate of the point that works for both lines. With this, we can now find the matching `x`-coordinate, by plugging in this value for `y` into either of our starting equations:

`"-"x+2color(red)y"   "=4`
`"-"x+2color(red)((3))=4`
`"-"x+6"     "=4`
`"-"x"            "="-"2`

`=>x=2`

That's it—we have found that the lines do cross, and the coordinates of the crossing point are `(x,y)=(2,3)`:

graph{(-x+2y-4)(5x-3y-1)=0 [-10, 10, -2, 8]}