Solving equations
What is the meaning of solving an equation? What is the meaning of finding roots of an equation? From a geometric perspective, if we have two squares equal to each other and one has a variable governing its side or area, solving the equation means to find the value of that variable such that both squares have the same area or same side. If the equation represents a function, it may be the case of finding the minimum distance, maximum energy or the instant of time at which the velocity changes its direction.
Did you know that some pretty common operations that we do when solving equations or systems of equations can be interpreted geometrically? Teachers that I had never made this relationship. 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. Think about this trivial exponential equation [math]\displaystyle{ a^x = a^4 \iff x = 4 }[/math]. If a number raised to some unknown power is equal to the same number raised to 4, what is the unknown? It can't be anything else than 4.
[math]\displaystyle{ a^2 + b^2 = c^2 }[/math] is a particular case and (almost) everyone knows the Pythagora's Theorem. Maybe this is behind people (erroneously) assuming that the sum of log is the log of the sum and the same for exp and square roots.
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 of an operation from one side to the other. We are doing the same operation on both sides at the same time.
There is another point of view regarding the zero. We often forget that any number minus itself is equal to zero. Therefore, we can add a number minus itself to any side of an equation without changing it. In the same way, a non-null number divided by itself is one and we can multiply anything by one without changing it. We often associate natural numbers with counting, such as 3 is 3 objects. But we can also see it as 3 x 1. That is, a rectangle with sides 3 and 1.
With linear systems we often do operations such as to multiply a line of the system by some constant. The concept that is behind such operation is a vector. A vector can be viewed as an arrow in 2D or 3D, with 4D and beyond being impossible to draw. Suppose that the equation represents a line, if we multiply every vector's coordinate by the same constant we keep the vector's orientation. We changed each coordinate by some constant rate. We didn't change the direction of the vector. If a line intersects another in space, changing the vector's magnitude or norm does not change the point of intersection. The same concept can be extended to any number of dimensions, we just won't be able to draw it in higher dimensions.
An analogy is to think on the principle of conservation of mass or the conservation of energy. As long as both sides of the equation are equal to each other, we aren't creating energy from nothing nor making mass disappear without a trace. A cube can have its volume split into two smaller cubes. We all learn at school that we can divide any quantity into any number of parts and the sum of the parts is equal to the original quantity. What we don't learn at school is to prove it. Going further into mathematics there are many types of equations for which we don't know if a solution exists or not. That's a whole different type of problem that is to find the solutions and under which conditions they exist. Or to prove that it's impossible to find integers that satisfy the equation.
