Question

explain the process of using graphing technology to solve a system of equations

Answer 1

The idea is this: we want a single point, (x, y), that satisfies both equations. Thus, we simply need to know which points (if any) are common to both functions, that is, where f(z) = g(z).

A way to solve a system of equations inspired by this insight can be seen in the following:

Given two equations,
ax + by = c
dx + ey = f

Let's rewrite these equations as functions of x.

ax + by = c => by = c - ax => y = f(x) = c/b - a/b*x
dx + ey = f => ey = f - dx => y = g(x) = f/e - d/e*x

We want the y's to be equal, thus we can solve for this:

f(x) = g(x) => c/b - a/b*x = f/e - d/e*x. Solving for x, we get:

d/e*x - a/b*x = f/e - c/b
x(d/e - a/b) = f/e - c/b
x = (f/e-c/b)/(d/e-a/b)

Once we have x, we have y. (Note: y = c/b - a/b*x). For a linear system of equations, it is possible to get a single solution, no solution, or an infinite number of solutions. That happens if d/e-a/b = 0, or in other words, when they have the same slope.

Visually, if you have the graph, you just need to see where the two lines intersect.