The intersections of 2 circles

Lets consider the following (random) question:

Find the intersections of the circles $c_1: x^2+y^2=25$ and $c_2: (x-2)^2 + (y-3)^2=9$

In order to solve this we can do $c_2-c_1$, which leaves us with $y=-\dfrac{4}{6}x+\dfrac{29}{6}$.

If we then substitute the $y$ into one of the circles, we get the intersections.

My question, a simple one, is: Why? What does the line $y=-\dfrac{4}{6}x+\dfrac{29}{6}$ represent? Why will we get the intersections if we substitute this particular line in the equation of the circles? Can someone give an intuitive explanation?

Pictures always help:

The line you obtain is derived, essentially, by expressing each circle as an equation equal to 0 and then equating the circles (which is essentially subtracting the equation of one circle from that of the other) to find the equation of the line $$\mathcal{l}: y=-\dfrac{4}{6}x+\dfrac{29}{6}$$ that passes through the points at which the circles intersect, say $p_1, p_2$. Since there are only two points of intersection, recall that two points define a line, so it is fitting that equation for $\mathcal{l}$ is a line connecting (and passing through) those points. (See the yellowish-brownish line.)

But that line itself doesn't tell us what those points of intersection are.

This line has infinitely many points; to determine which of those points are actually points that the both circles have in common (i.e., to determine the actual points of intersection), we use the equation of the line $\mathcal{l}: y = $ expressed as a function of $x$, and substitute $y$ into either circle's equation. Doing so, we obtain the points on the circles that lie on $\mathcal{l}$ and thus obtain the points $p_1, p_2$where $c_1, c_2$ intersect.

Suppose we have some kind of equations $c_1$ and $c_2$ for arbitrary shapes (now circles), using the $2$ coordinate variables $x,y$. Then, if $P=(x_0,y_0)$ is a common point on the two shapes, it means exactly that $c_1(x_0,y_0)$ and $c_2(x_0,y_0)$ are valid statements, consequently so are their sum or any linear combination: $\alpha c_1+\beta c_2$ for $\alpha,\beta\in\Bbb R$.

Now, geometrically what do these linear combinations represent in the original case of circles? It is going to be almost always a circle, but, if $\alpha+\beta=0$, it becomes the equation of a line. But this line still shares the common points of $c_1$ and $c_2$ by the above remark.

Then, what you called 'substitue the line' in one of the original equations, say $c_1$, will give the inteersections of the line and the circle: basically it is just solving the given system of equations.

See also this picture.