5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Students should be able to understand how manipulating systems of equations will lead to those equations' solutions. Let's say we have two functions that we're trying to solve simultaneously. We'll call our functions f(x, y)and g(x, y), because they're funky and groovy. Yeah, we took our functions straight out of the 70s.

For instance, let's say that f(x, y) is 2x + 4 = 3y and g(x, y) is -x + 8y = 2. We can rewrite these as 2x – 3y + 4 = 0 and x – 8y + 2 = 0, in which case f(x, y) = 2x – 3y + 4 and g(x, y) = x – 8y + 2. Now, we have our awesome functions (or…funk-tions?).

If one solution to both f(x, y) = 0 and g(x, y) = 0 is when x = a and y = b, then f(a, b) = 0 and g(a, b) = 0. That means a and b must also be the x and y values for the system of equations f(x, y) + g(x, y) = 0 and nf(x, y) = 0, where n is a constant. After all, f(a, b) + g(a, b) = 0 + 0 = 0, and nf(a, b) = n(0) = 0.

Basically, this means that system of equations f(x, y) = 0 and g(x, y) = 0 will have the same solutions as f(x, y) + g(x, y) = 0 and nf(x, y) = 0.

Students should know that this is useful when solving a system of equations (as in, 2 or more of 'em) because it means we can multiply equations by constants and add them together all we want. That'll help us find the x and y values that work for all equations in the system. 

We strongly urge you to relate this information back to graphing linear and quadratic equations. Explain that the solution to the system of equations is really the point (or points) where the functions intersect. For example, functions f(x, y) and g(x, y) intersect at (a, b) since those are the x and y values that make the two equations equal to each other. It's all connected, just like we're all connected. Pretty groovy, right?