Systems Of Linear Equations and Linear Systems

Systems Of Linear Equations and Linear Systems.md

Linear Equations and Systems of Linear Equations And Their Applications

Linear Equations

The Composition of A linear Equation

constants, variables, coefficients

Constants

Constants are unchanging and typically represented by either a real number or an abstraction like pie ($\pi$) or perhaps another equation like $e=mc^2$.

For instance, in the equation $Y = MX + B$ $m$ $M$ and $B$ are constants. This equation is in slope intercept form and give us a way to convey and reason about seemingly complex things in a seaming simplistic manner. The word seemingly is used because complexity and simplicity are objective and inseparable from the assumptions, understandings, and possibly misgivings of the student or practitioner.

All that being said, when broken down and reasoned about the equation of $Y = MX + B$ and any other equation can be broken down into the relationship of constants to everything else, in this case, other parts of the equation.

Variables

Variable are place holders for real numbers or abstractions that are, in the case of $Y = MX + B$ are X and Y. M and B are constants or constant abstractions. Y and X are variables that change depending on where each of them exist in space and or time.

This "existence" is defined by what is called a point. A point in space and or time. This means that X and Y, our variables, are related to, equal to, one another based on the effects their constants, M and B have on them.

Coefficients

Coefficients are denoted by their relation. For instance in the term $xy$, $x$ has a coefficient of $y$ and $y$ has a coefficient of $x$. This is a fancy way of saying, when and if you ever find out what $x$ and or $y$ are, multiply them by one another. For example, in the term $xy$ if $y$ is = 33 our term becomes $x*33$, which also can be written as $x(33)$.

So in $y = mx + b$ if m were equal to, lets say 5, then our equation would effectively be $y = 5(x)+b$, which also can be written $$y = 5 * x + b$$

If we look at the different writings together we can clearly see the benefit of "abstracting" away the multiplication symbol.

$$y = 5 * x + b$$ $$y = 5x + b$$

But, when we have a value for our variable, if the constant and the coefficient are both real numbers we add either parenthesis or the multiplication symbol to clearly define the relationship between variable and coefficient in an instance where we know what value will "occupy" the space of the variable. For example, where $x$ is equal to $33$ we would write our equation in one of these two ways.

$$y = 5 * 33 + b$$ $$y = 5(33) + b$$

Time and Space

Time and Space are depicted by what is known as the cartesian coordinate system. The system consists of two lines. One line is called the "y-axis" the other line is called the $x-axis$.

The $y$ axis represents a vertical plane, a line that is straight up and down, north to south, that divides negative and positive numbers. The center of this line is Zero. Vertical lines have a Slope, M value, that is undefined. Not Zero, Undefined because zero cannot be divided by zero.

The $x$ axis represents a horizontal plane, a line that is strait from left to right, right to left, east to west, west to east, that divides negative and positive numbers. Horizontal lines have a Slope, M value, that is 0. Zero denotes that there is no vertical, $x-axis$ change over time. Things are flat.

To graph is to put a point on this line. $(x,y)$. $x$ tells us where our point is on the $x-axis$, the horizontal plane. $y$ tells us how far above or below the horizontal plane ,$y=axis$, a point is.

When we have two or more points on space, and plot them on a coordinate system such as the cartesian coordinate system we are able to reason about the relationship between those two points.

Linear Equations are way to reason about the relationship between two points. One form if a linear equation is the Slope Intercept form.

Slope Intercept Form - Y = MX + B and Its Applications.

Intercepts and Slope

Slope intercept form expresses key information in a simple form, but only if the "formula" is understood. $y=mx+b$ While these letters represent numbers, meaning any of them could potentially be any number, in this form M and B gives us all the information we need to make an infinite number of calculations, recursions, regressions, and projections.

M

$m$ represents Slope. $Slope$ represents change. In order for change to occur there must be be a plane of some sort, an expanse, a divide, a starting point and an end point.

$Slope$ is defined as $\frac {rise}{run}$ (rise over run), which is the rate of change over distance, time, etc...

If there is no expanse, no distance, no time, etc.. If there is no $run$ there is no rate of change. $\frac{rise}{0}$ or "Slope" undefined because there is no expanse to define it. The conclusion is a vertical line.

Conversely, if there is a divide, a distance or time greater than or less than zero the result is a line with a "slope". If rise is zero, $\frac{0}{run}$ the result is a line that is horizontal, flat.

$\frac{rise}{run}$ is yet another abstraction. Rise is derived from defining the relationship between two point, just at a different resolution. $\frac{rise}{run} = \frac{y - y_{1}}{x - x_{1}}$, the tiny $_{1}$ is subscript, which in this case indicates that we are to subtract the $x$ and $y$ variables of one point from another point.

For instance if we hade two points, or $(1, -2)$ and $(2,-1)$ our, Slope, $\frac{rise}{run}$ , would $=$ $\frac{-2 - (-1)}{1 - 2}$.

B

$b$ is what is called the $y-intercept$. The $y-intercept$ is the point at which a line crosses a vertical line known as the $y-axis$.

X

$x$ can be derived from solving the equation. This means that by replacing the right variables with the right numbers we are able to predict where something will be in time or space.

$y = mx + b$ or its principles have been used by man throughout time immemorial to build everything that has ever been built, and has been used to explore everything that has ever been explored, from atoms and their behavior to the alleged expansion of the universe. Even more so, this equation can be found in absolutely everything. It has been used to build great monuments, nations, and fortunes.

For example the equation $Y = $200X + 0$ where is a time frame, will tell us how much money we will accumulate if we save $200$ at a time.

Systems Of Linear Equations and Linear Systems

According to the Merriam-Webster Dictionary,

A system is a regularly interacting or interdependent group of items forming a unified whole. System. (n.d.). Retrieved December 14, 2020, from https://www.merriam-webster.com/dictionary/system

The whole is the result. A system is a whole things composed of smaller things that share some sort of relationship.

A linear equation could be referred to as a system of constants, variables and coefficients thus,

Multiple linear equations that have a relationship with one another are referred to as a System of Linear Equations

If a linear equation gives us a way to calculate lines, systems of linear equations give us a way to calculate and configure multiple lines. This is magnificent in the sense that we do not have to describe all of the calculations. We can simply present a list of equations.

Since a linear equations like in the form of $Y = MX + B$ give us change over space and time we can map relationships between these changes. This is used in financial markets to projects profit and loss and by actuaries to predict risk, with each equation being a risk or reward factor. All together these equations for a system that overtime and space can indicate causation, correlation, and produce new equations to be investigated.

Physicists use sets of equations to map and explore the universe, rocket scientist use them to map and explore propulsion, navigation, life support, and various other systems. Architects use them to design fanciful works of architecture, structure engineers use a different set of equations bases on and derived from materials, weather patterns and various other "points" of data to make sure these magnificently designed structures don't fall over.

Where equations and sets of equations have proven to be the most prolific to date is in the design, production, manufacture, and distribution of goods and services via the internet and computer systems.

A large complex system of infinite possibilities perched atop a simple equation $Y = MX + B$. The power of this understanding is immeasurable.