Problem 58 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 10 x+5[6-(2 x+3)] $$

Step-by-Step Solution

Answer

Our simplified result is \(15\).

Step 1: Simplify the Expression Inside the Brackets

We start by simplifying the bracketed expression. That gives us: \(10x + 5[6-2x-3]\).

Step 2: Simplify the Expression Inside the Brackets Further

Next, we need to subtract the last two terms inside brackets. This gives us: \(10x + 5[3 - 2x]\).

Step 3: Distribute the 5 across the terms inside the brackets

We distribute the term 5 across the inside of the brackets by multiplying it with each term in the brackets. This gives us: \(10x + 5*3 - 5*2x\).

Step 4: Simplify Further

After performing the multiplication from the previous step, we get: \(10x + 15 - 10x\).

Step 5: Combine Like Terms

The last step is to combine like terms. That is, we add the terms involving 'x' together, and subtract the terms involving 'x' from the numbers. This gives us: \(0 + 15\).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, combining like terms is a crucial step. Like terms are terms that have the same variables raised to the same power. These can be combined to simplify expressions.

• For example, if you have two terms like \(10x\) and \(-10x\), these can be combined because they have the same variable, 'x', raised to the same power.
• In our problem, \(10x - 10x\) creates \(0\), effectively removing the 'x' term altogether.
• Even without a variable, constant numbers are considered like terms and can be combined. In our example, the constant terms \(15\) and \(0\) were the result of combining like terms, leading to a simplified result of \(15\).

This step ultimately reduces the expression, making it more understandable and manageable. Remember, combining like terms is all about recognizing the commonality between terms and then performing the necessary arithmetic operations.

Distributive Property

The distributive property is used to simplify expressions by distributing multiplication over addition or subtraction within parentheses. In simpler terms, you multiply the term outside the parentheses with each term inside the parentheses.

• In the equation \(5[3 - 2x]\), the distributive property directs us to multiply 5 by each term in the brackets. This results in \(5 \times 3\) and \(5 \times -2x\).
• This simplification results in \(15 - 10x\).

Using the distributive property allows you to remove parentheses effectively and makes it easier to spot like terms in the expression. It can often look complicated at first, but breaking it down into smaller steps, by distributing one part at a time, helps in understanding the effect of each operation.

Removing Grouping Symbols

Grouping symbols such as parentheses \(()\), brackets \([]\), and braces \(\{\}\) are often used in algebra to dictate the order of operations. Removing these symbols is a key part of simplifying algebraic expressions.

• In the initial expression, \(10x + 5[6-(2x+3)]\), the brackets indicate that the subtraction \((6-(2x + 3))\) should be resolved first.
• This simplifies to \(10x + 5[3 - 2x]\) after handling the operations within the brackets.
• The subsequent use of the distributive property further removes the brackets, giving us a clearer expression \(15 - 10x\).

Removing grouping symbols correctly relies on following the order of operations diligently. This step is critical in ensuring that you simplify the expression appropriately and thoroughly, so the final result is as concise as possible.