Problem 84 - Elementary Algebra Within Reach

Question

In Exercises \(75-86\), simplify the expression. $$ 5(x+9)-2(30+4 x) $$

Step-by-Step Solution

Answer

The simplified expression is \(-3x - 15\).

Step 1: Expand the Brackets

When expanding brackets in \(5(x+9)-2(30+4x)\), we multiply each term within the brackets by the factor outside. For the first bracket, we get \(5x + 45\). For the second bracket, we get \(-60 - 8x\). So the expression becomes \(5x + 45 - 60 - 8x\).

Step 2: Group Like Terms

In order to simplify the expression, we group like terms together. We group the terms with x together and the constant terms together. This gives us \((5x - 8x) + (45-60)\).

Step 3: Simplify The Expression

Now that we've grouped the like terms, simply subtract: for terms with x, we have \(5x - 8x\) which simplifies to \(-3x\). For the constants, we have \(45 - 60\) which simplifies to \(-15\). Therefore, our final simplified expression is \(-3x - 15\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their simplest form, where no further simplification is possible. This process makes complex expressions more manageable and easier to work with. Let's break it down:

- **Expand Parentheses:** Start by distributing numbers outside parentheses (explained in detail in the next section) to remove the parentheses and obtain a straightforward expression.
- **Combine Like Terms:** Identify and group similar types of terms together to further simplify the expression.

Simplification reduces errors when solving equations and understanding the core of the problem. Simplified expressions are tidier and provide clearer insight into the relationships between variables and constants.

Distributive Property

The distributive property is a fundamental algebraic principle that helps us eliminate parentheses in expressions and make calculations easier. It states that multiplying a number by a sum is the same as multiplying each addend by the number outside the brackets. For example, in the expression \(5(x + 9)\):

- Multiply each term inside the parentheses by 5.
- This gives us the products \(5x\) and \(45\), leading to the expression \(5x + 45\).

This property is crucial for simplifying expressions because it moves us towards combining like terms. It transforms terms under a single operation thereby reducing complexity. Understanding the distributive property allows you to handle more complex algebraic expressions with confidence.

Combining Like Terms

Once we have expanded the brackets using the distributive property, the next step is to combine like terms. This step further simplifies the expression without changing its value.

Like terms in an expression are terms that have the same variable raised to the same power. For example, the terms \(5x\) and \(-8x\) are like terms because they both include the variable \(x\) raised to the first power.

- Subtract the coefficients of like terms (here, \(5x - 8x\) results in \(-3x\)).
- Separate the constants. Compute their difference (\(45 - 60\)) which gives \(-15\).

By combining these terms, the original expression \(5x + 45 - 60 - 8x\) is reduced to \(-3x - 15\). This makes it much clearer and easier to understand. Mastering this concept is vital in algebra because it streamlines problem-solving and helps identify the most significant components within mathematical expressions.