Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\(x+(y+10)=(x+y)+10\)

Step-by-Step Solution

Answer

The equation \(x+(y+10)=(x+y)+10\) illustrates the associative property of addition.

Step 1: Analyze the Equation

The equation given is \(x+(y+10)=(x+y)+10\), which is of the same form as the associative property of addition. In mathematical form, the associative rule for addition states: \(a+(b+c) = (a+b)+c\). Hence the equation appears to be displaying this rule.

Step 2: Compare the Equation with the Rule

When comparing \(x + (y + 10)\) and \((x + y) + 10\) to \(a + (b + c)\) and \((a + b) + c\), it's clear that \(x\) aligns with \(a\), \(y\) aligns with \(b\), and \(10\) aligns with \(c\). Therefore, the equation aligns with the associative property of addition.

Step 3: Conclusion

Given the equation matches the associative property for addition rule, it is confirmed that this equation is illustrating the said rule.

Key Concepts

Rules of Algebra

When diving into algebra, one of the first things you'll learn are the "rules of algebra". These rules help us simplify expressions and equations accurately. They ensure consistency in calculations across different problems.

• Commutative Property: This rule states that the order of adding or multiplying numbers does not affect the result. For example, in addition, it says \(a + b = b + a\).
• Associative Property: This is what our original exercise is about. It refers to the grouping of numbers. It states that no matter how numbers are grouped in addition or multiplication, the result will be the same. For addition, it is expressed as \(a + (b + c) = (a + b) + c\).
• Distributive Property: This rule connects addition and multiplication. It is expressed as \(a \cdot (b + c) = a \cdot b + a \cdot c\).

These fundamental properties or rules are applied extensively to various algebraic problems, aiding in simplifying and solving equations efficiently.

Addition in Algebra

Addition in algebra can seem a bit more abstract than simple number addition. However, it's just as straightforward once you grasp the concept.
In algebra, we often use letters to represent numbers—these are called variables. The most important thing to remember is that the rules of addition still apply. For example:

• \(a + b\) is the sum of variables \(a\) and \(b\).
• We can also add constants, like numbers, to variables: \(a + 5\) where 5 is a constant.

The associative property is often used in addition to simplify complex expressions. If you see expressions like \(x + (y + z)\), you can freely regroup them as \((x + y) + z\) without altering the result. This flexibility can be especially handy when solving equations or reorganizing terms.

Basic Algebra Concepts

Starting with basic algebra concepts is key to mastering more advanced topics. Here are a few things you should get comfortable with:

• Variables and Constants: Variables represent unknown values, while constants are fixed numbers. They are the building blocks of algebraic expressions.
• Expressions and Equations: An expression is a combination of variables, constants, and operators like addition. An equation, on the other hand, shows equality between two expressions.
• Simplifying Expressions: This involves using the rules of algebra to make expressions as concise as possible. For example, combining like terms or using properties like associative and commutative to rearrange terms.

Understanding these core concepts makes tackling algebra problems much easier and lays a strong foundation for future learning. Remember, practice is key—working through various examples will help these concepts become second nature.