Problem 36 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 6(3 x+4) $$

Step-by-Step Solution

Answer

Question: Multiply the expression (3x + 4) by 6 and simplify. Answer: 18x + 24

Step 1: Identify the expression and operator

Identify the expression to be multiplied by 6 and the operator in the problem. In this case, the expression is \((3x + 4)\) and the operator is the multiplication by 6.

Step 2: Use Distributive Property

Use the distributive property of multiplication by applying the operator to each term inside the parentheses. In this case, it would be: $$ 6 \cdot (3x + 4) = 6 \cdot 3x + 6 \cdot 4 $$

Step 3: Perform Multiplication

Perform the multiplication of each term: $$ 6 \cdot 3x + 6 \cdot 4 = 18x + 24 $$

Step 4: Combine Like Terms (if any)

In this case, there are no like terms to be combined. So, the final simplified expression is: $$ 18x + 24 $$

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is all about making them easier to work with by breaking them down into their simplest forms. Imagine you're trying to clean your room; you want to organize things so that every item is in its right place. Simplifying an algebraic expression involves doing the same. You want to process the mathematics into a neat form. To start, identify any terms that have the same variable factor and combine them. Then, handle any operations like addition, subtraction, multiplication, or division inside the expression precisely.
In our exercise, the simplification process uses the distributive property to distribute the multiplication across the terms within the parentheses. Breaking down complex looking expressions into manageable batches is a great way to simplify matters. The ultimate goal is to ensure the expression looks its best and ready to compute or further process.

• Simplify terms by using properties of arithmetic operations.
• Make sure you've accounted for all like terms by the end.

Combining Like Terms

Combining like terms is an essential part of simplifying algebraic expressions. It involves adding or subtracting terms with the same variable and exponent. Think of like terms as matching pairs of socks in your drawer—they need to be put together. By combining like terms, you consolidate the expression, making it simpler and more streamlined.
For the expression provided, \(18x + 24\), if there were terms like \(10x\) or \(12x\), they could be summed up with \(18x\) to form a new expression. Since none exist here, the expression stays the same. This step is crucial when dealing with more complex expressions, ensuring clarity and simplifying the solving process.

• Match terms with the same variables and exponents.
• Add or subtract coefficients of like terms.
• Remove any redundant terms.

Algebraic Multiplication

Algebraic multiplication is quite similar to the typical multiplication you're familiar with, but it involves letters (variables) as well as numbers (coefficients). When you are asked to perform algebraic multiplication, you are distributing a factor across every term within a set of parentheses. The factor is usually outside the parentheses, and you multiply it with each term inside.
In our exercise, you see how 6 multiplies with every term inside the parentheses: \(3x + 4\). You end up multiplying 6 with both \(3x\) and \(4\), leading to the products \(18x\) and \(24\) separately. Finally, you add these results together to get the final expression \(18x + 24\).

• Apply the multiplicative factor to each term inside the parentheses.
• Multiply both numerical coefficients and variable terms.
• Organize results before proceeding to simplify.