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

Feb 9, 2015

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:
$x + 2 y = 4$ and $- x - 3 y = - 7$

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 $x = 4 - 2 y$
and
$- x - 3 y = - 7$ is re-written as $x = 7 - 3 y$

Since both right-hand sides are equal to x, we have:
$4 - 2 y = 7 - 3 y$
Adding $\left(+ 3 y\right)$ 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
$x = 7 - 3 y$ substituting 3 for y gives $x = 7 - 3 \cdot 3$ or $x = 7 - 9$
Therefore $x = - 2$

And we have the solution:
$\left(x , y\right) = \left(- 2 , 3\right)$