Problem 18 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \(2+(8+y)\)

Step-by-Step Solution

Answer

The expression simplifies to \(10 + y\) using the associative property.

Step 1: Understand the Associative Property

Remember that the associative property allows us to regroup numbers in an addition or multiplication problem without changing the result. Specifically, for addition, it states that \( (a + b) + c = a + (b + c) \).

Step 2: Apply the Associative Property

In the expression \( 2 + (8 + y) \), we can use the associative property to regroup the terms. According to the property, we can change this to \( (2 + 8) + y \).

Step 3: Simplify the Expression

Now, simplify the expression \( (2 + 8) + y \). Calculate the sum inside the parentheses: \( 2 + 8 = 10 \). Substitute the result back into the expression to simplify it further: \( 10 + y \).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols like plus or minus. The fundamental part of understanding algebraic expressions lies in recognizing how these different elements interact. For example, in the expression \(2 + (8 + y)\), there are both numbers, such as \(2\) and \(8\), and a variable, \(y\).
Variables represent unknown values and are often used to generalize mathematical statements. Here, \(y\) can be replaced by different numeric values.

• **Constants**: Fixed numbers, like \(2\) and \(8\) in this example.
• **Variables**: Symbols representing unspecified numbers, such as \(y\).
• **Operators**: Indicate calculations, such as plus signs \(+\).

Understanding these components will help you manipulate and simplify expressions effectively.

Simplifying Expressions

Simplifying an expression involves reducing it to its simplest form without changing its value. This process can make expressions easier to work with or solve.
Look at \((2 + 8) + y\). Here, we simplify part of the expression by performing the arithmetic operation inside the parentheses first. Calculate the sum: \(2 + 8 = 10\). This makes the expression simpler: \(10 + y\).

• **Combine Like Terms**: Simplify expressions by adding or subtracting numbers and variables when possible.
• **Arithmetic Simplification**: Always perform any operations inside parentheses first.

After simplifying, you're left with an expression that is easier to understand and work with: \(10 + y\). This expression is now in its simplest form.

Properties of Addition

One major property of addition used in algebra is the associative property. This property tells us that no matter how we group the numbers when adding, the result will remain the same.
For example, in \(2 + (8 + y)\), using the associative property allows us to "regroup" the numbers as \((2 + 8) + y\). This rearranging doesn’t change the outcome, which simplifies the work needed to find a solution.

• **Associative Property**: Rearranging the parentheses in an addition operation does not change the sum.
• **Usage**: Especially useful in simplifying complex expressions like \(a + (b + c)\) into \((a + b) + c\).

It's critical to understand that the associative property is about how numbers are grouped, not the sequence of the numbers themselves. This basic principle simplifies problem-solving in algebra by allowing more flexibility in computation.