Question

What is the point of intersection of the lines `x+2y=4` and `-x-3y=-7`?

Answer 1

As Realyn has said the point of intersection is `x=-2, y=3`

"The point of intersection" of two equations is the point (in this case in the xy-plane) where the lines represented by the two equations intersect; because it is a point on both lines, it is a valid solution pair for both equations. In other words, it is a solution to both equations; in this case it is a solution to both:
`x + 2y = 4` and `-x - 3y = -7`

The simplest thing to do is to convert each of these expressions into the form `x =` something
So `x + 2 y = 4` is re-written as `x = 4 - 2y`
and
`-x - 3y = -7` is re-written as `x = 7 - 3y`

Since both right-hand sides are equal to x, we have:
`4 - 2y = 7 - 3y`
Adding `(+3y)` to both sides and then subtracting `4` from both sides we get:
`y = 3`

We can then insert this back into one of our equations for x (it doesn't matter which), for example
`x = 7 -3y` substituting 3 for y gives `x = 7 - 3*3` or `x = 7 -9`
Therefore `x = -2`

And we have the solution:
`(x,y) = (-2,3)`