Problem 16 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 5(x-1)+7(x+4) $$

Step-by-Step Solution

Answer

The simplified expression is \(12x + 23\).

Step 1: Distribute the Constants

First, distribute the constants outside the parentheses to each term inside them. For the expression, distribute 5 and 7 to their respective terms:- Distribute 5 to each term in \((x-1)\): \[5(x-1) = 5 \cdot x - 5 \cdot 1 = 5x - 5\]- Distribute 7 to each term in \((x+4)\): \[7(x+4) = 7 \cdot x + 7 \cdot 4 = 7x + 28\]This gives us the expression: \[5x - 5 + 7x + 28\]

Step 2: Combine Like Terms

Combine the like terms in the expression. Look for terms that have the same variable:- Combine the \(x\) terms:\[5x + 7x = 12x\]- Combine the constant terms:\[-5 + 28 = 23\]So the simplified expression is:\[12x + 23\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify expressions. It involves distributing a multiplication operation over an addition or subtraction within parentheses. This means that each term inside the parentheses is multiplied by the factor outside. For example, in the expression \(5(x-1)\), you multiply 5 by each term inside the parentheses: \(5 \cdot x - 5 \cdot 1 = 5x - 5\). The same rule applies to the other part of our example, \(7(x+4)\), where 7 is multiplied by both \(x\) and 4, giving \(7x + 28\).

The distributive property not only simplifies algebraic expressions but also prepares them for further simplification, such as combining like terms. This makes it easier to handle complex equations and is a critical step in algebra.

Combining Like Terms

Once you've used the distributive property to remove the parentheses, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our exercise, after distribution, we end up with the expression: \(5x - 5 + 7x + 28\).

To combine like terms, you simply add or subtract the coefficients of the terms that are identical other than their numeric component. In our case:

• Combine the \(x\) terms: \(5x + 7x = 12x\).
• Combine the constant terms: \(-5 + 28 = 23\).

This process reduces the expression to \(12x + 23\), where 12 is the coefficient of the \(x\) term, and 23 is the remaining constant.

Combining like terms is a way to further simplify expressions and make them easier to understand or solve.

Simplifying Expressions

Simplifying expressions is the process of transforming complex algebraic expressions into a more manageable form. By using both the distributive property and combining like terms, you can take an expression like \(5(x-1) + 7(x+4)\) and break it down into something simpler.

After utilizing the distributive property, the expression becomes \(5x - 5 + 7x + 28\). The next step is combining like terms, which leads to the simpler expression \(12x + 23\).

Simplified expressions are much easier to work with and are especially useful when solving equations or substituting variables. Achieving a simplified form allows for clearer insight into the relationships between terms and better manipulation for further mathematical operations. Simplification is a vital skill that enhances mathematical problem-solving abilities.