Problem 27 - Elementary and Intermediate Algebra

Question

Use the associative law of addition to write an equivalent expression. $$ (a+5)+b $$

Step-by-Step Solution

Answer

The equivalent expression is \( a + (5 + b) \).

Step 1 - Identify the parts of the expression

In the given expression \( (a + 5) + b \), the parts inside the parentheses are \(a + 5 \), and this is added to \(b \).

Step 2 - Apply the associative law of addition

The associative law of addition states that the way in which numbers are grouped when adding does not change their sum. It can be expressed as \[ (x + y) + z = x + (y + z) \].

Step 3 - Re-group the expression using the associative law

Using the associative law, we can re-group the original expression \( (a + 5) + b \) as \( a + (5 + b) \). This shows that the sum remains the same despite the re-grouping.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (like letters), and arithmetic operations (like addition and multiplication). For example, in the expression \( (a + 5) + b \), \( a \) and \( b \) are variables, and we use addition to combine them with the number \( 5 \). These expressions can be simplified or re-grouped using various mathematical rules.

Algebraic expressions are used to represent real-world situations mathematically, such as calculating the perimeter of a shape or solving for unknown quantities. Understanding how to manipulate these expressions is crucial in algebra, and applying laws like the associative law helps us simplify and solve problems more easily.

When dealing with algebraic expressions, always look for opportunities to use properties and laws to make your calculations simpler and more straightforward.

Properties of Addition

Addition has several important properties that help us simplify and solve equations. One key property is the associative property, which is central to our exercise. The associative property states that the way in which numbers are grouped when adding does not change their sum. It can be written as \((x + y) + z = x + (y + z)\).

Other important properties include:

• Commutative Property: This property says that the order of addition does not affect the sum. For example, \a + b = b + a\.
• Identity Property: Adding zero to any number does not change the value of that number. For example, \a + 0 = a\.

These properties make it easier to work with algebraic expressions and simplify them. By understanding and applying these properties, we can solve complex problems more efficiently.

Grouping in Mathematics

Grouping in mathematics refers to the concept of using parentheses to alter the order in which operations are performed in an expression. This is especially relevant when applying properties like the associative law.

In our example \( (a + 5) + b \), we initially group \( a \) and \( 5 \) together. By using the associative property of addition, we can re-group the expression as \ a + (5 + b) \. Both expressions will give the same result, but re-grouping can make calculations or further manipulations easier.

When working with more complex equations, proper grouping can significantly simplify the solving process. It helps ensure that operations are performed in the correct order, and often reveals simpler or more efficient ways to approach a problem. Always pay attention to grouping to avoid errors and make your work easier.