Problem 77 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$-5(2 x)$$

Step-by-Step Solution

Answer

The simplified expression is \(-10x\).

Step 1: Identify the operation

In the expression \(-5(2x)\), the operation signified is multiplication.

Step 2: Use the distributive property

Distribute the \(-5\) across the \(2x\). In other words, multiply \(-5\) by each term inside the parentheses.

Step 3: Multiply the numbers

Perform the multiplication. This gives us \(-5*2x\), which simplifies to \(-10x\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form while preserving the original meaning. This process allows you to more easily understand and work with the expression afterward.
To simplify an algebraic expression, follow these general steps:

• Identify all the terms and operations involved in the expression.
• Follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Combine like terms. Like terms have the same variable raised to the same power. For example, combine terms like \(3x\) and \(5x\) to get \(8x\).
• Simplify any remaining operations, such as reducing fractions or simplifying complex numbers.

In our specific example, \(-5(2x)\), using these steps allows us to see quickly that the expression reduces to \(-10x\). It shows a direct multiplication rather than any combination of like terms since it's a simple one-term multiplication problem.

Distributive Property

The distributive property is a fundamental concept in algebra that is used to eliminate parentheses and simplify expressions. It states that a term outside the parentheses can be multiplied by each term inside the parentheses.
In mathematical terms, the distributive property is expressed as: \[a(b + c) = ab + ac\]. This means that if you have a number multiplied by a sum or difference, you must multiply the outside number by each term within the parentheses.
In the expression \(-5(2x)\), applying the distributive property is a straightforward way to eliminate the parentheses:

• Distribute \(-5\) through the \(2x\), giving you \(-5 imes 2x\).
• It simply converts the expression from a multiplication problem with parentheses into a direct multiplication problem without them.

This property is especially useful when dealing with more complex expressions, helping you break down and recombine terms methodically.

Multiplication in Algebra

Multiplication in algebra might seem more complicated than regular arithmetic multiplication, but once understood, it becomes quite simple. Algebra involves multiplying coefficients (the numerical parts of terms) and variables, if present.
Here are the steps to multiply in algebra:

• Multiply the numerical coefficients. For instance, in \(-5(2x)\), we would multiply \(-5\) and \(2\) to get \(-10\).
• Apply the result to the variable(s) in question. The variable itself remains unchanged unless multiplied by another instance of the same variable. Here, there is only \(x\), so the variable part is unchanged, resulting in \(-10x\).
• If multiple variables are involved, distribute the multiplication across each, maintaining their exponents. Remember that \(x^1\) times \(x^1\) would become \(x^2\).

By understanding these concepts, you gain the clarity needed to tackle more complex problems, as seen in simplifying expressions that involve both coefficients and variables.