91Ó°ÊÓ

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms within a parenthesis. Specifically, it states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products of that number with each individual term inside the parenthesis. This property can be expressed as:

• For addition: \(a(b+c) = ab + ac\)
• For subtraction: \(a(b-c) = ab - ac\)

When applying this to our expression \(5(x-0.5)\), we multiply 5 with each term inside the parenthesis individually. Thus, \(5 \times x - 5 \times 0.5 = 5x - 2.5\). The same process is repeated for each group like \(0.3(x-2)\) and \(-0.7(x+7)\), resulting in \(0.3x - 0.6\) and \(-0.7x - 4.9\) respectively.
Mastering the distributive property will greatly ease the process of working with complex algebraic expressions.

Combining like terms is the process of simplifying algebraic expressions by adding or subtracting terms that have the same variable and exponent. This is an essential skill for simplifying expressions and solving equations.
In our exercise, after applying the distributive property, we are left with the expression \(5x - 2.5 + 0.3x - 0.6 - 0.7x - 4.9\). Here, we have like terms which can be combined:

• The terms with the variable \(x\): \(5x + 0.3x - 0.7x\) are combined to become \(4.6x\)
• The constant terms: \(-2.5 - 0.6 - 4.9\), which simplify to \(-8.0\)

By grouping together the terms with the same variable and performing simple arithmetic, students can effectively combine like terms to streamline expressions. This makes the final equation or expression easier to interpret and solve.

Simplifying an expression involves reducing it to its simplest form by making all operations possible, combining like terms, and removing any unnecessary parentheses. The goal is to make the expression as clean and straightforward as possible.
After distributing and combining like terms in the exercise, we end up with the expression \(4.6x - 8.0\). This is now considered simplified because:

• All possible operations have been carried out
• There are no more like terms to combine
• The expression is free of parentheses

Simplifying expressions helps reveal the relationship between different variables and constants more clearly and allows for easier manipulation and comparison of expressions. It's a powerful tool in both algebraic problem-solving and real-world applications.