Linear Equations Comprehensive Study Essay

Concepts of Linear Equations

Linear equations are functions of the form y=mx +b written in equation notation or f(x)= MX + b in function notation; y=mx +b is known as the standard form. These equations are referred to as the equation of a line and they have many applications in the field of mathematics. Some specific applications include area of the triangle given as = a x b, simple interest formulas I = p x r x t, and velocity given by the formula distance/time, v/t.

In these notations m and b are fixed values; m determines how y changes for every single unit of change in x with this change in x (Δx) units the result is a corresponding change in y (Δy = mΔx units). Hence, we can obtain m by solving Δy/Δx while b has a vital role in the equation for instance when x=0, the equation notation will be y=b. (Hodgkin). Linear equations can be solved through other elementary operations such as addition, subtraction, multiplication, and division. However, these operations are limited to the number of variables which when increased may make them complex to solve.

Notable Inventors of this Concept

Rene Descartes is one among many mathematicians who invented and worked extensively on linear equations; he was the first to coin the term linear to mean something that is in a row or a line and identify the functions surrounding this mathematical concept (Reilly). Rene Descartes was instrumental in developing the field of mathematics that pertains to linear equations and specifically the equation of a line. Rene refined the idea of the standard notation equation of a line given as y=mx + b, m which represents the slope and determines the change in y for every unit change in x, b is a constant determined by y-intercept (Reilly).

By using two intersect equations i.e. (x/a) +(y/b) =1, Rene was able to show that linear equations can be solved graphically by plotting different values of y against x values and obtaining corresponding values of y when x=0, y-intersect, and x when y=0, x-intersect. For instance, through Rene’s ideas, we realize that if a linear equation is plotted on a graph the results are a straight line that has a slope of this line being m. This observation is constant for all straight lines and would enable solving of linear equations just by picking the values of y-intersect and x-intersect as the solution of the equation.

Evolution of Linear Equations

Now we can take this concept of a linear equation to another level and use it to solve linear models by use of point-slope formula. For example, equation of a line passing through the point (x₁, y₁) which has slope m will be given by y=mx-b, where b=y₁ – mx_1. This equation can be used to find the equation of a line when information of a point and slope of a line is given or where y-intercept is given. Linear equations for horizontal and vertical lines given the point (x₁, y₁) will be represented as y = y₁ and x = x₁ respectively. This means an equation of the line through points (-1, 7) which has slope -3 will be given by y = -3x + b_. Consequently, an equation of the horizontal line through (-1, 7) is given by y = 7. While an equation of the vertical line through the same point is given by x = -1.

Thus, linear equations can be used in solving models and in predicting data by applying the same concepts that we have discussed above. For instance, with a given collection of data points such as (x₁, y₁),…, (x_n, y_n).

These values y₁, y₂,…, y_n will be the observed values. By modeling these data with a linear equation y=mx +b we can substitute given values of x in our equation and the resulting values of y will be the predicted y values. Linear equations are very important in solving equations where there is more than one unknown variable that needs to be determined. Using a similar related mathematical application of the Gaussian elimination method developed by Gaussian we can also effectively solve linear equations through addition elimination (Burton). The Gaussian elimination method is a process that involves back-substitution, for example, the solution for the following linear equation will be given by:

Z+2y-x=1

Z+2x=3

Z=1

By substituting the value of z in equation 2 we get x=1 which allows the solution of y to be found in equation 1 as y=1/2.

Works Cited

Burton, D. “The History of Mathematics: an Introduction.” Columbus: McGraw-Hill Company.2010. Print.

Reilly, A. “The History of Linear Equations, 2011.” Web.

Hodgkin, H. L. “A history of Mathematics. from Mesopotamia to modernity.” New York: Oxford university press. 2005. Print.