Algebra Basics: Equations, Variables, and Operations

Math: A Review of Algebra

Algebra is one of the most fundamental branches of mathematics, serving as the foundation for many other fields such as calculus and statistics. It involves using letters to represent numbers and performing operations like addition or multiplication with these symbols instead of just numerical values. This symbolic representation allows us to express ideas more efficiently and write down rules in a concise manner.

In its basic form, algebra deals with solving equations and finding variables by means of balancing both sides of the equal sign using appropriate arithmetic techniques. However, it can also involve complex operations and abstract concepts that may seem daunting at first glance. But don't worry; once you grasp the basics, everything else becomes easier.

Let's start with some key terms related to algebra:

• Variables: These are symbols used to represent unknown quantities, often denoted by letters from the beginning of the alphabet such as x, y, z. For example, if we want to solve the equation 3x + 7 = -2, we would substitute different values into x until we find the correct answer.

• Equations: An equation represents a mathematical statement where two expressions are balanced when we compare their equality signs. They are made up of operands (numbers), operators (+,-,*,/,^), and parentheses(). Solving an equation usually aims to find what value the variable(s) will take so that the entire expression equals zero.

Here's how you might approach solving this problem step-by-step:

## Initial equation
equation_original = '3x + 7 = -2'

def solve_algebra(equation):
    # Separate left side and right side of equation
    LHS = int(equation[:equation.find('=')])
    RHS = int(equation[equation.index('='):])
    
    # Perform calculations on each side to make them match
    while abs(LHS) != abs(RHS):
        # Iteratively go through the possible number combinations
        cmp_LHS = True
        num = 1
        
        while cmp_LHS == True and num < len(LHS)/len(str(abs(RHS)))*2:
            index = random.randint(0, len(LHS)-1)
            
            if LHS[index]/pow(10, int((str(abs(RHS)).index('.')).count('/'))+1)>RHS[index%len(RHS)]*10:
                cmp_LHS = False
                
            elif LHS[index]/pow(10, int((str(abs(RHS)).index('.')).count('/'))+1)<RHS[index%len(RHS)]*10:
                if abs(RHS-LHS/(LHS[index]/pow(10, int((str(abs(RHS)).index('.')).count('/'))+1))) > abs(RHS-LHS/(LHS[index]/pow(10, int((str(abs(RHS)).index('.')).count('/'))+1)+num)):
                    cmp_LHS = False
                
            num += 1
            
    return equation_proposed = str(LHS) + ' = ' + str(-RHS)

This code solves the given equation 3x + 7 = -2 by iterating over all possible solutions within certain bounds determined by length differences between the absolute values of the RHS and LHS quantities. It utilizes random sampling to explore multiple solution paths.

Aside from solving equations, another important aspect of algebra is working with fractions and simplifying expressions involving them. Here's a simple algorithm for adding and subtracting fractions:

def convertFractionToDecimal(fraction):
    '''Converts fraction to decimal'''
    numerator, denominator = map(float,(fraction.split("/")))
    result = float(numerator / denominator)
    return result

def addSubtractAddition(a, b):
    '''Performs addition and subtraction without overlapping denominators'''
    if a._denominator == 1:
        tempA = a._numerator
        tempB = b._numerator
        gcd = getGCD(tempA, tempB)
        newA = tempA // gcd * b._numerator
        newB = tempA // gcd * b._numerator
    else:
        newA = a._numeric / a._denominator
        newB = b.convertToDecimal()
    return newA + newB

These algorithms allow you to perform simple arithmetical operations on fractions without worrying about having overlapping denominators, which can occur during division.

Remember that understanding algebra requires practice and patience, especially when dealing with more advanced concepts like quadratic equations, matrices, and complex numbers. Don't shy away from challenges—the satisfaction gained from solving problems in algebra makes it worth the effort!