Solving equations

Teachers that I had never made this relationship. Did you know that some pretty common operations that we do when solving equations or systems of equations can be interpreted geometrically? Very often we are taught that an equation that has squares, roots, powers, trigonometric terms, are non-linear and that's it. All we are taught is that non-linear systems or equations are more complicated to solve than linear cases.

Suppose that we have [math]\displaystyle{ a + b = c }[/math]. The most common interpretation for this is that we are adding up two numbers that are equal to a third. What if each side of the equation represents the side of a square? How many times did you see or did you do it yourself the operation to calculate the square on both sides? That operation is nothing more than to assume that each side represents the side of a square. If they are both equal, then the areas of both squares should be equal as well. We can naturally extend it to cubes and higher dimensions. That explains, geometrically, why doing operations such as taking the log, the square or the square root on both sides doesn't change the equality.

I think there is a common confusion that seems to arise from an apparent contradiction. [math]\displaystyle{ a + b = c \iff a = c - b }[/math]. We inverted the sign of [math]\displaystyle{ b }[/math] and moved it from the left to the right side of the equation. If both sides of the equation are equal to each other, shouldn't we erase from one side and copy it to the other side without changing the operation? There is the confusion! The previous operation is really this one: [math]\displaystyle{ a + b = c \iff a + b - b = c - b }[/math]. It seems that every teacher of physics, linear algebra, numerical methods and calculus sees this mistake very often. We aren't really inverting a sign or an operation from one side to the other. We are doing the same operation on both sides at the same time.

Now we all learn that [math]\displaystyle{ |x| \ge 0 }[/math] and [math]\displaystyle{ x^2 \ge 0 }[/math]. Are they the same? No. They are both always positive. But modulus is about the length or distance, whereas the square is about the area. There are many cases in different scenarios where we see the operation to use the absolute value or the square. To use one or the other requires some distinction between what is a distance and what is an area. It really depends on the problem itself. Both the absolute value and the square yield functions that have symmetry because negative values are reflected. There are two differences: the absolute value creates a point where the function is continuous but non-differentiable; the other difference is that the square calculates the square and by doing so it changes the rate of change.