Algebra 1 - Methods for Solving Equations & Matrix Applications

This Algebra 1 assignment explores different methods for solving linear systems of equations, with a focus on the substitution method, comparing it to graphing and elimination. The student explains the process, advantages, and provides a convincing argument for its superiority. The assignment also covers matrix multiplication, explaining the conditions for multiplication and what the results represent with an example. Finally, the assignment delves into real-world applications of matrices, including their use in Google search algorithms, computer graphics, message encryption, and robotics, providing citations to support the claims. The student demonstrates an understanding of the concepts and their practical relevance.

Running head: ALGEBRA 1
Algebra
Name of Student
Institution
Date

ALGEBRA 2
Question 1.
 I prefer solving the linear system of equations using the method of substitution. This
method entails plugging one value of an isolated variable to the other equation. The new
equation carrying only one variable is then solved. This method is simple and quick to
use.
 Substitution method is better than the other two because it is a fast way of solving the
equations and does not involve complex calculations. Compared to the elimination
method, the latter is also simple but sometimes the method involves extra calculations to
find factors that when multiplied with the two equations the coefficients of one variable
becomes isolated. The graphing method, on the other hand, is useful only when the
solutions are whole numbers (Bapat, 2010). It becomes difficult to read answers from the
graphs whenever the solutions are fractions or decimals.
 Substitution method is the easiest to understand. Given a system of linear equations, all
that is required is to make one of the variables isolated then substitute its solution to the
other equation. From there you can simply find the solutions.
Question 2.
A matrix is composed of rows and columns. Multiplication of two matrices requires that
the number of columns of the first matrix is equivalent to the number of rows of the
second matrix. For example, a 3 by 2 matrix can be multiplied by a 2 by 4 matrix. The
resulting matrix will be an n by m matrix. The number of rows of the first matrix gives n
while m is the number of columns of the second matrix. In our example above the
resulting matrix will be three rows by four columns.

ALGEBRA 3
The result of matrices multiplication indicate that the product represents composition of
linear maps. Multiplication of matrices is the basic tool of linear algebra, having several
applications in numerous areas of mathematics. Whenever two linear maps are
represented by matrices, the composition of the two maps is represented by the matrix
product. Consider the following example;
5=3 x −4 y
11=x + y
the linear composition can be found via matrix multiplication as follows:
3 −4
1 1
x
y= 5
11
The solution to this is the linear composition.
Question 3.
The following are main real-world application of matrices (Horadam, 2012):
 In google search the stochastic matrices and eigen vectors help algorithms ranking used
in ranking pages of the web.
 Matrices assist in projecting a 3-dimensional image into a 2-dimensional screen, in
computer-based applications, thus forming the most real seeming movements. Each
matrix representing 2d images in computers contain a unique value of integers which
represent the level of brightness or associated properties.
 Matrices assist in the encrypting message codes. Programmers use matrices and their
inverse for message coding or encryption. Message is formed as a binary format for
passing information and follows code theory for solving. Matrices help to solve those
equations.

ALGEBRA 4
 Matrices are the base elements in robotics and automation. Robot movements are
programmed by calculating rows and columns of matrices. The execution for controlling
the robots are formed based on matrices calculation.
 In physics and engineering, matrices are applied in various calculations. Electronics,
airplane and spacecraft, networks, and chemical manufacturing all require complex
calculations from matrix transformations.

ALGEBRA 5
References
Bapat, R. B. (2010). Graphs and matrices (Vol. 27). London: Springer.
Horadam, K. J. (2012). Hadamard matrices and their applications. Princeton university
press.
MacDuffee, C. C. (2012). The theory of matrices (Vol. 5). Springer Science & Business
Media.