ORIGINAL BOOK: Athanasios Paraskevopoulos, Dashel Myers, Exploring Algebra with Wolfram Language: A High School Textbook, ISBN: 979-8867299187.Kobo:https://www.kobo.com/gr/en/ebook/exploring-algebra-with-wolfram-language-a-high-school-textbook AMAZON: https://a.co/d/5OxLJIq.
I am excited to share with you my latest project, "Exploring Algebra with Wolfram Language: A High School Textbook." This comprehensive guide is designed to help high school students master algebraic concepts using the powerful Wolfram Language. To give you a sneak peek into the content, I' ve prepared a notebook with a section from the book.
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I’m excited to announce that the new version of this book includes chapters contributed by Dashel Myers, a high school student and Wolfram Student Ambassador. Dashel approached these chapters with a level of professionalism that adds a unique perspective to the material, making it even more engaging and relevant for students. I believe this collaborative effort enhances the overall educational experience and brings a fresh, student-centric approach to the exploration of algebra using the Wolfram Language.
Linear and Non-Linear Systems
In this chapter, we will explore the world of linear and non-linear equations. We will start by understanding what equations are and how they are formed. We will then delve into the different types of equations, namely linear and non-linear equations, and learn how to solve them. We will also learn about systems of equations and how to solve them using various methods. The chapter will be filled with examples and exercises to help you understand and apply the concepts better. So, let’s embark on this mathematical journey!

Linear Equations

Let's start with the following definition.

Definition

A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one - degree equation . The standard form of a linear equation in one variable is of the form
Ax+B0
.
Here,
x
is a variable,
A
is a coefficient and
B
is constant . The standard form of a linear equation in two variables is of the form
Ax+ByC
. Here,
x
and $
y
$ are variables,
A
and
B
are coefficients and
C
$
is a constant.
There are linear equations in one variable

Example

x + 7 = 12 ,
5
2
x-91
, and
x
3
+5
x
2
-3
are equations in one variable
x
.
and linear equations in two variables.

Example

2x+3y15
,
7x-
y
3
3
are equations in two variables
x
and
y
.
The graph of a linear equation always forms a straight line.

Linear System Of Equations

In this chapter, we will quickly review what a
22
system of equations is, and then we will look at how to apply these systems to real - world problems . We will see how to set up a system to correspond to a problem and discuss solutions.

22
System of Equations

A system of equations is a set of two or more equations with the same variables . In this particular example, there are two equations and two variables, therefore, we call this a
22
system of equations.

Example

Let' s say we are given the following example :
3x+5y-2
​
​
2x-7y9
This representation is a system of 2 linear equations with two unknowns. We call the solution of the system the values that the variables must take so that they simultaneously verify the two equations. For example, in the above system, the solution is
x1,y-1
The solutions of a system of equations are the values of the variables that make all the equations true . A solution of a system of two linear equations is represented by an ordered pair
(x,y)
.
To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation . If the ordered pair makes both equations true, it is a solution to the system .

Methods for Solving a 2x2 linear systems

Fortunately, there are many ways you can use to solve two - by - two systems, and you have the benefit of choosing which method to use . The most commonly used methods are :
◼
  • Graphing
    • ◼
  • Substitution
    • ◼
  • Elimination
    • The graphing method is based on, no surprise, graphing the two equations and trying to visually determine where these two lines intersect (if they intersect at all) . This method is naturally limited to approximations in most cases .

    How to Solve a system of linear equations by graphing.

    1
    .
    Graph the first equation.
    2
    .
    Graph the second equation on the same rectangular coordinate system.
    3
    .
    Determine whether the lines intersect, are parallel, or are the same line.
    4
    .
    Identify the solution to the system.
    5
    .
    Check the solution in both equations.
    The Substitution method is based on the idea that one can solve for one variable in one of the equations, and then plug that into the other equation, to solve for the other variable . Often times this is convenient, because the structure of one of the equations may make it direct to solve for one variable . However this is not always the case, and this method is largely limited to the case of
    22
    systems .

    How to Solve a system of linear equations by substitution.

    1
    .
    Solve one of the equations for either variable.
    2
    .
    Substitute the expression from Step 1 into the other equation.
    3
    .
    Solve the resulting equation.
    4
    .
    Substitute the solution in Step 3 into either of the original equations to find the other variable.
    5
    .
    Write the solution as an ordered pair.
    6
    .
    Check that the ordered pair is a solution to both original equations.
    Let' s solve another example
    One hundred and thirty - six tourists were transported in 10 buses of 12 and 16 seats . How many buses were there of each type?
    We denote by
    x,y
    the required values and construct the equations of the system.
    Let
    x
    be the number of 12-seat buses and
    y
    the number of 16-seat buses.
    We’ve got the system:
    x
    +
    y
    10
    12
    x
    +16
    y
    136
    ⟺
    x
    +
    y
    10
    3
    x
    +4
    y
    34
    ​⟺
    x
    10-
    y
    30-3
    y
    +4
    y
    34
    ⟺
    x
    6
    y
    4
    A solution to a system of equations consists of values of the variables that make all of the equations in the system true, and the set of all solutions to a system is called the solution set of the system . Consider our example above . If we plug in
    x6
    and y = 4 into each of the equations, both of the equations are true .
    In the Mathematica code, the above system will be written with the Solve command as follows :
    In[]:=
    sys={x+y==10,12x+16y==136};​​sol=Solve[sys,{x,y}]
    Out[]=
    {{x6,y4}}
    To solve the system graphically we plot the lines
    (
    ϵ
    1
    )
    and
    (
    ϵ
    2
    )
    , represented by the two equations, on a common axis system and consider the coordinates of their common points as the solution of the system .
    In particular, we have the following cases :
    ◼
  • When the lines
    (
    ϵ
    1
    )
    and
    (
    ϵ
    2
    )
    intersect, then the system has a unique solution. This solution is the coordinates of their intersection point
    M(
    x
    0
    ,
    y
    0
    )
    • ◼
  • When the lines
    (
    ϵ
    1
    )
    and
    (
    ϵ
    2
    )
    are parallel, then the system has no solution. We say that the system is unsolvable.
    • ◼
  • When the lines
    (
    ϵ
    1
    )
    and
    (
    ϵ
    2
    )
    are identical, then they have infinite common points, namely all points on the common line, i.e. the system has infinite solutions. We then say that the system is indeterminate.
    • So, we can solve the system graphically with the following command :
    In[]:=
    Plot[{-x+10,-(3/4)x+17/2},{x,-10,10},PlotLegends->"Expressions"]
    Out[]=
    -x+10
    1
    4
    (-3)x+
    17
    2
    Which of the above three categories does the problem we just solved belong to, based on the graph?

    Solving 3x3 Systems of Linear Equations

    In this section, we will learn how to solve systems of equations that have three equations and three variables .

    What Is a 3x3 System of Linear Equations?

    To define a
    33
    system of linear equations we need to understand what each part of the term means . Linear equations form straight lines when they are graphed . They have a degree of one, meaning that the variables have no more than one exponent . A system of equations has two or more equations that are solved simultaneously . When a system of equations is
    33
    , it has three equations and three variables . The goal of solving a system of equations is to find a value for each of the variables that satisfy all of the equations . In a
    33
    system of linear equations, we need to find a value for each of the three variables that make each equation true .

    Solving by Substitution

    In the example, we see how an expression from one equation can be substituted for a variable in another equation . The goal is to have an equation with one variable that we can solve for . Once we find the value of one variable, we can use it to solve for the others .
    Solve the following system
    2x-y+w-5
    ​​
    y+3w7
    ​​
    w2
    So the solution of the original system is the triad
    (-3,1,2
    ) .
    In[]:=
    sys={2x-y+w==-5,y+3w==7,w==2};​​sol=Solve[sys,{x,y,w}]​​Plot3D[{-5-2x+y,(7/3)-(1/3)y,2},{x,-4,4},{y,-4,4},PlotLegends->"Expressions"]
    Out[]=
    {{x-3,y1,w2}}
    Out[]=
    -5-2x+y
    7
    3
    -
    y
    3
    2
    Since the solution of a linear system
    33
    , as we saw above, is reducible to the solution of a linear system
    22
    , it follows that a linear system
    33
    either has a unique solution or is unsolvable or has an infinite number of solutions .

    Computer Exercises

    1
    .
    Solve the following system of equations
    ​
    2x+5y-1
    ​
    ​
    -10x-25y5
    ​
    and sketch the graph .
    2
    .
    Translate to a system of equations and then solve:
    When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for 20 minutes, her fitness app said she burned 278 calories. When she spent 20 minutes on the elliptical trainer and 30 minutes on circuit training she burned 473 calories. How many calories does she burn for each minute on the elliptical trainer? How many calories for each minute of circuit training?
    3
    .
    Solve the following system of equations
    x-2y-z-1
    ​
    -x+2y-3z-4
    ​
    ​
    3x-2y-7z0

    Historical Note

    The use of the word "algebra" for denoting a part of mathematics dates probably from the 16 th century . [citation needed] The word is derived from the Arabic word al - jabr that appears in the title of the treatise Al - Kitab al - muhtasar fi hisab al - gabr wa - l - muqabala (The Compendious Book on Calculation by Completion and Balancing), written circa 820 by Al - Kwarizmi .
    Al - jabr referred to a method for transforming equations by subtracting like terms from both sides or passing one term from one side to the other, after changing its sign .
    Therefore, algebra referred originally to the manipulation of equations, and, by extension, to the theory of equations. This is still what historians of mathematics generally mean by algebra.
    In mathematics, the meaning of algebra has evolved after the introduction by François Viète of symbols (variables) for denoting unknown or incompletely specified numbers, and the resulting use of the mathematical notation for equations and formulas . So, algebra became essentially the study of the action of operations on expressions involving variables . This includes but is not limited to the theory of equations .
    At the beginning of the 20th century, algebra evolved further by considering operations that act not only on numbers but also on elements of so - called mathematical structures such as groups, fields, and vector spaces . This new algebra was called modern algebra by van der Waerden in his eponymous treatise, whose name has been changed to Algebra in later editions .

    CITE THIS NOTEBOOK

    Explore Algebra with Wolfram Language: New High School Textbook Available!​
    by Athanasios Paraskevopoulos​
    Wolfram Community, STAFF PICKS, November 18, 2023
    ​https://community.wolfram.com/groups/-/m/t/3067312