Problem 49 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ -(5 x+2) $$

Step-by-Step Solution

Answer

The expression \( -(5x+2) \) simplifies to \( -5x - 2 \).

Step 1: Understanding the Distributive Property

The distributive property is a rule in algebra which states that multiplying a single term by terms inside parentheses is equal to multiplying each term inside the parentheses separately by the single term. The rule can be expressed as \( a(b + c) = ab + ac \). In this exercise, our task is to apply the distributive property to the expression \( -(5x+2) \).

Step 2: Apply the Distributive Property

Apply the distributive property to the expression \( -(5x+2) \). This means we need to multiply both terms inside the parenthesis by -1:- \( -1 imes 5x = -5x \)- \( -1 imes 2 = -2 \)After distributing, the expression becomes \( -5x - 2 \).

Step 3: Simplify the Expression

After applying the distributive property, we check if the expression \( -5x - 2 \) can be further simplified. In this case, \( -5x \) and \( -2 \) are not like terms, so the expression is already in its simplest form.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities. This makes it easier to form equations and solve problems that contain unknown values. Algebra allows us to work with variables, which are symbols used to represent unknown or changeable values.
The key advantage of algebra is its ability to generalize arithmetic. Instead of solving specific numbers each time, algebra gives us tools to solve problems with patterns and rules.

• It helps us understand the relationships between different mathematical quantities.
• It allows us to create formulas that can be repeatedly used.

Many real-world problems can be solved using algebra, making it a valuable subject to master. In exercises like the one we're discussing, we often deal with expressions that need simplification or evaluation. Knowing the basic rules and concepts of algebra can significantly aid this process.

Simplification Made Simple

Simplification in algebra refers to the process of making an expression easier to understand or work with, without changing its value. It often involves combining like terms and using algebraic rules, such as the distributive property.
When we simplify expressions, we are aiming to write them as compactly as possible. This often makes calculations more straightforward:

• Combine terms that have the same variable and power.
• Use arithmetic operations to perform any possible calculations.

In the exercise example, after applying the distributive property, the expression becomes \( -5x - 2 \). Checking for like terms (terms that have the same variables and powers) shows us that no further simplification was possible. Thus, the expression is already as simple as it can be.

Power of Parentheses

Parentheses in algebra play an essential role in determining the order of operations and grouping terms. They indicate that the operations within them should be performed first. However, when using the distributive property, they need to be opened by multiplying each term inside them by the term outside.
This process of removing parentheses by distributing is crucial for simplification:

• Helps in organizing terms clearly.
• Ensures operations inside are prioritized and clearly grouped.

In the given exercise, the expression \( -(5x+2) \) includes parentheses, and by using the distributive property, we effectively "distribute" the -1 to both inside terms. This step shows how parentheses can group terms that need to be carefully handled during calculations.

Handling Like Terms

Like terms are terms in algebraic expressions that have the same variable raised to the same power. Combining like terms is a crucial step in simplifying expressions, as it reduces complexity by adding or subtracting coefficients.
To identify like terms, look at their variables and exponents:

• Terms must have exactly the same variable part to be combined.
• Only the coefficients (the numbers multiplying the variables) can be added or subtracted.

In our example \( -5x - 2 \), there are no like terms because one is a term involving \( x \) while the other is a constant. Recognizing this is important because it shows that the expression is already simplified and further combining isn't possible.