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.

[math]\displaystyle{ a^2 + b^2 = c^2 }[/math] is a particular case and 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.

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.

At school we all learn that 1 kg can be divided in (0.5 + 0.5) kg. One bottle of water can be divided in as many glasses as we want. Some glasses may be taller, others larger, or many glasses equal to each other. What is important is that we cannot create water out of nowhere. The sum of parts cannot be larger than the original quantity. That's the principle of conservation of mass in chemistry. Or the conservation of energy in physics. Can the volume of a cube be divided into two equal and smaller cubes? Sure it can if we allow for fractions and irrational numbers. [math]\displaystyle{ a^3 + b^3 = c^3 }[/math]. In daily life we can have two swimming pools such as the sum of their volumes is a cube. But are the two smaller pools cubes? Most of the time they aren't perfect cubes and we don't need them to be anyway. [math]\displaystyle{ a^n + b^n = c^n }[/math]. Now this generalisation for n dimensions lose any visual interpretation. Taking the log on both sides, multiplying by a constant or adding a constant to both sides won't help us prove anything here. The mathematics required to find all the numbers that may solve that equation, or if there aren't any numbers to solve it, is not seen at undergraduate levels.