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Algebra is a crucial branch of mathematics that deals with symbols and the rules for manipulating those symbols. It enables us to create expressions and equations, which we can then solve to find unknown values. The foundation of algebra involves understanding how numbers and variables interact with each other. This is done through various properties and operations.

In algebra, we often use letters to represent numbers. These letters or symbols allow us to create generalized equations. For example, in the expression \(5(4+3)\), each part of the equation can be represented by a letter, like \(a(b+c)\). This flexibility means we can apply algebraic principles to wide-ranging problems, giving us a powerful tool for solving real-world issues.

Simplifying expressions is an essential skill in algebra that helps make complex problems more manageable. The process involves several steps, such as combining like terms and using mathematical properties. The goal of simplification is to express an equation or expression in its simplest form.

• Combining Like Terms: This means adding or subtracting terms that have the same variable and exponent. For example, in the expression \(20 + 15\), there are no variables involved, thus we can simply add the constants together to simplify to \(35\).
• Use of Properties: Utilizing properties like the distributive property allows us to simplify expressions by transforming them into a form that is easier to handle, thus helping to see the results more clearly.

Simplifying expressions not only makes solving problems easier but also aids in better understanding the underlying mathematical structure.

Arithmetic operations are fundamental mathematical processes that involve adding, subtracting, multiplying, or dividing numbers. In algebra, these operations are applied to both numbers and algebraic expressions, allowing us to formulate solutions to equations.

In the given exercise, we used the distributive property, which is an example of how multiplication is used in algebraic expressions. The property itself takes the form \(a(b+c) = ab + ac\). It essentially allows us to "distribute" the multiplication over addition inside the parentheses.

• Addition: After applying the distributive property and multiplying through, the expression becomes \(20 + 15\), which involves a straightforward addition.
• Multiplication: This was a key step in the process, as it simplified the expression from \(5 \times 7\) to \(35\).

Understanding these operations and how they apply to algebraic principles is vital for solving and simplifying complex mathematical problems.