Problem 45 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 5(x+2)-(3 x-4) $$

Step-by-Step Solution

Answer

The simplified expression is \(2x + 14\).

Step 1: Distribute the Coefficient

Apply the distributive property to remove the parentheses by distributing the 5 and -1 through the expressions. Start with the first term: \(5(x+2) = 5 \cdot x + 5 \cdot 2 = 5x + 10\).

Step 2: Distribute the Coefficient (Continued)

Now distribute the -1 into the second expression inside the parentheses: \(-1(3x - 4) = -1 \cdot 3x - 1 \cdot (-4) = -3x + 4\).

Step 3: Combine Like Terms

Now combine like terms from both expressions. From \(5x + 10 - 3x + 4\), combine \(5x\) and \(-3x\) to get \(2x\), and \(10\) and \(4\) to get \(14\).

Step 4: Write the Simplified Expression

Combine the simplified terms to write the final expression: \(2x + 14\).

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra that allows us to remove parentheses and simplify expressions. It states that a term multiplied by a sum or difference inside parentheses can be distributed to each individual term within the parentheses. This can be written as:

• \( a(b + c) = ab + ac \)
• \( a(b - c) = ab - ac \)

In the given exercise, we first encounter the expression \(5(x+2)\). Applying the distributive property, we multiply 5 with each term inside the parentheses:

• \(5 \cdot x = 5x\)
• \(5 \cdot 2 = 10\)

This gives us \(5x + 10\). Similarly, for the expression \(-(3x-4)\), we use -1 as the coefficient and distribute it:

• \(-1 \cdot 3x = -3x\)
• \(-1 \cdot (-4) = 4\)

Thus, the expression becomes \(-3x + 4\). This step is crucial for breaking down expressions into manageable pieces.

Combining Like Terms

After applying the distributive property and getting rid of the parentheses, we find ourselves with several terms that belong to the same category, such as those with the same variable or constant terms. These are called 'like terms,' and by combining them, we simplify the expression further.
In the expression from the exercise \(5x + 10 - 3x + 4\), we spot the like terms easily:

• \(5x\) and \(-3x\) are like terms (both include \(x\)).
• \(10\) and \(4\) are like terms (both are constants).

By combining the \(x\) terms, we calculate:

• \(5x - 3x = 2x\)

And for the constants, we do:

• \(10 + 4 = 14\)

Bringing these together, we simplify the expression to \(2x + 14\). Combining like terms reduces complexity and aids in solving equations.

Simplifying Expressions

The ultimate goal of working with algebraic expressions is to simplify them as much as possible. Simplifying makes it easier to understand and solve problems by reducing the expression to its simplest form.
Through distributing and combining like terms, we transformed the original expression \(5(x+2)-(3x-4)\) to its simplest form \(2x + 14\). The process:

• Ensures there are no parentheses.
• Reduces the expression to a single expression without excess or unnecessary terms.

When simplifying, always aim for clarity. An expression like \(2x + 14\) is more straightforward and ready for further mathematical operations than its original form. This skill is particularly useful when solving equations or verifying solutions.