Problem 94 - College Algebra

Question

Simplify algebraic expression. \(6-5[8-(2 y-4)]\)

Step-by-Step Solution

Answer

The simplified expression is \(10y - 54\).

Step 1: Expand Brackets

We first tackle the internal brackets: \(2y - 4\). But since there's no operation to be performed in them, we leave them as is. Next is to expand the external brackets remembering that the minus sign before a bracket affects all terms within the bracket. Thus, \(5[8-(2 y-4)]\) becomes \(5[8 - 2y + 4] = 5 * 12 - 5 * 2y = 60 - 10y\)

Step 2: Simplify Expression

Now we substitute this back into the original equation to get: \(6 - (60 - 10y)\). Remembering again the effect of the negative sign before a bracket we get: \(6 - 60 + 10y = 10y - 54\)

Step 3: Final Simplified Expression

The final simplified expression is \(10y - 54\)

Key Concepts

Distribution in Algebra

Distribution is a key tool in algebra that allows you to simplify expressions by removing parentheses. Think of distribution as **spreading out** everything inside a bracket by multiplying.

• Start by identifying the term outside the bracket; here it's the number or variable you will multiply inside.
• In our original problem, the number 5 needs to be distributed through the expression in the brackets.
• This means you multiply 5 by each term in the expression inside the bracket, which involves both numerical and variable terms.

This concept is crucial for simplifying expressions because it breaks them down into smaller, manageable parts, aiding further calculation. Distribution is much like distributing cookies evenly to a group of children, ensuring each gets what they deserve!

Order of Operations

The order of operations is like the traffic rules for mathematics. It guides you on which operations to do first so everyone gets the same correct answer.

• Parentheses or brackets are always the first to be tackled in any algebraic expression.
• In our example with the expression \(6-5[8-(2 y-4)]\), you'd start by assessing the innermost brackets \((2y - 4)\), even though no further simplification was needed for that part.
• Next, you solve any operations in brackets before multiplying by terms outside the brackets due to distribution.
• Continue with multiplication and division (from left to right) before moving on to addition and subtraction.

By following these rules, you ensure your final expression is correct every time.

Negative Sign Distribution

The negative sign is an important part to consider, especially when placed outside a bracket. It acts like a coat of paint changing the sign of every term inside the bracket.

• When we had \(5[8-(2 y - 4)]\), the sequence changes to \(5[8 - 2y + 4]\), applying the minus sign inside.
• When distributing a negative sign, each term is essentially multiplied by \(-1\). Thus, it flips all the signs of the terms inside the bracket.
• This concept helps maintain the balance of an expression and prevents errors that might arise from ignoring subtraction as distribution happens.

With practice, you'll come to see a negative sign distribution as an automatic process, just like flipping a pancake!