Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the expression. \(4 x+x(13-7)\)

Step-by-Step Solution

Answer

The simplified expression is \( 10x \).

Step 1: Distribute the Multiplication

Take the expression inside the parentheses and distribute it to the variable outside. The expression is \( x(13 - 7) \). Multiply \( x \) by each term inside the parentheses: \( x \times 13 \) and \( x \times (-7) \). This yields: \( 4x + 13x - 7x \).

Step 2: Combine Like Terms

Now combine the like terms \( 4x \), \( 13x \), and \( -7x \). Adding these gives: \( (4 + 13 - 7)x = 10x \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify expressions involving multiplication over addition or subtraction. It states that when you have a term outside of parentheses, you can distribute it across each term inside the parentheses. In algebraic terms, this means:

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

Let's look at a practical example with our exercise: with the expression \( x(13 - 7) \), we apply the distributive property:

• Multiply \( x \) with 13, giving you \( 13x \).
• Then, multiply \( x \) with -7, giving \( -7x \).

Putting it all together, the distributed expression becomes \( 13x - 7x \). Don't forget to carry along any other terms outside the parentheses, such as the original \( 4x \) from our expression, leading to \( 4x + 13x - 7x \). This lays the foundation for the next step of simplifying the expression further.

Combining Like Terms

Combining like terms is a crucial simplification step that makes algebraic expressions more manageable. Like terms are terms that have the same variables raised to the same power. Only coefficients of like terms can be combined. For example, in **4x** and **13x**, the variable part is the same (**x**), so they can be added or subtracted from each other.To combine like terms effectively, follow these easy steps:

• Check for terms that have the same variable component. In our expression: \( 4x, 13x, -7x \) are all like terms because they all have \( x \) as the variable.
• Add or subtract the coefficients. Here, you compute:\( 4 + 13 - 7 = 10 \).

Therefore, the expression simplifies to \( 10x \). Now, everything with the same variable is grouped together neatly. Combining like terms is like tidying up your room; it makes everything much more organized and easier to handle.

Expression Simplification

Expression simplification is the process of breaking down complex algebraic expressions into their simplest form. This often involves applying various algebraic rules, such as the distributive property and combining like terms, to make an expression easier to understand or solve.By using expression simplification techniques, you achieve three primary goals:

• Reduce the complexity of the expression.
• Make mathematical operations faster and less error-prone.
• Prepare the expression for solving equations or further manipulation.

In the exercise \( 4x + x(13 - 7) \), we simplified it step by step:

• First, applied the distributive property to get \( 4x + 13x - 7x \).
• Then, combined like terms to reach \( 10x \).

The final result is a streamlined expression that's ready for any further mathematical tasks you may need. Remember, simplification is your best friend when dealing with algebraic expressions—it saves time and helps prevent mistakes.