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Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. These symbols represent numbers and variables, and algebra allows us to solve equations and find unknown values.
In this case, we're dealing with an equation from the Pythagorean theorem, a foundational concept in algebra and geometry. It's expressed as \( c^2 = a^2 + b^2 \), which describes the relationship between the sides of a right triangle. Algebra helps us rearrange this formula to solve for one of the sides, specifically \( a^2 \) in this case.
When we rearrange the equation using algebraic principles, we are manipulating the symbols to isolate the variable of interest. This skill of transforming equations is a fundamental part of algebra, which allows us to solve for unknowns efficiently.

Solving equations is a core aspect of algebra where the goal is to find the value of the unknown variables that satisfy a given equation.
For the Pythagorean theorem equation \( c^2 = a^2 + b^2 \), we're aiming to isolate \( a^2 \). This means performing operations that get \( a^2 \) by itself on one side. We start with subtracting \( b^2 \) from both sides of the equation to move terms around.

• First, we have \( c^2 = a^2 + b^2 \).
• Subtract \( b^2 \) from both sides, leading to \( c^2 - b^2 = a^2 \).
• The result, \( a^2 = c^2 - b^2 \), is what we were solving for.

This simple process of moving terms to isolate the variable is key in solving equations. By ensuring the variable of interest is alone on one side of the equation, we can more easily find its value.

Variables are symbols used in mathematics to represent numbers or values that can change. They are essential for performing algebraic operations and solving equations. In the exercise, variables are \( a, b, \) and \( c \). Each stands for a length of a side of a right triangle, showcasing their dynamic nature in real-world problems.
Variables allow us to form relationships between different quantities, reflected in our equation \( c^2 = a^2 + b^2 \). Here, \( c \) is the hypotenuse, while \( a \) and \( b \) are the other two sides. Understanding what each variable represents helps in knowing what we're solving for.
In solving for \( a^2 \), we treat \( a \) as a variable whose value we determine by manipulating the equation. This demonstrates the power of variables in expressing mathematical ideas and solving complex problems by representing unknown but changeable quantities.