Problem 68 - College Algebra

Question

simplify each algebraic expression. $$ 6-5[8-(2 y-4)] $$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression is \(10y - 34\).

Step 1: Simplify Inside the Brackets

Following the BODMAS order of operation, begin by simplifying the operation inside the bracket first. The expression inside the brackets is \(2y - 4\). As there are no operations to simplify, move to the next step.

Step 2: Distribute the -5 to the Terms Inside Brackets

Distribute or 'expand' the -5 to each term inside the square brackets. This gives \(6 -5*(8 - (2y - 4)) = 6 - 5*8 + 5*(2y - 4)\). After multiplication, this simplifies to \(6 - 40 + 10y - 20\).

Step 3: Simplify the Resulting Expression

Combine the like terms in the resulting expression from step 2 to simplify it further. The given expression now simplifies to \(6 - 40 + 10y - 20\) which leads to \(-34 + 10y\). This can be written in a better form as \(10y - 34\).

Key Concepts

BODMAS order of operations

The BODMAS rule is your guide to solving mathematical expressions in the correct order. BODMAS stands for:

• Brackets
• Orders (i.e., powers and square roots, etc.)
• Division and Multiplication (left to right)
• Addition and Subtraction (left to right)

To solve an expression correctly, always start with operations inside the brackets before moving on to orders, and then work through division/multiplication and addition/subtraction from left to right.
In our exercise, we follow the BODMAS rule by tackling the inside of the brackets first, even if no simplification is needed. Begin by focusing on smaller expressions within any kind of parentheses, which sets the stage for accurately simplifying the rest of the expression.

Distributive property

The distributive property allows us to multiply a term across a sum or difference inside a set of brackets. This property helps in simplifying expressions and solving equations.
Mathematically, it states: \[ a(b + c) = ab + ac \]This rule helps spread the factor outside the brackets through each term within.
In the exercise, the distributive property is used to multiply \[-5\] across the terms inside the brackets: \[ 8 - (2y - 4) \].
By distributing \[-5\], we rewrite the expression as \[ -5*(8) + 5*(2y - 4) \]. We then calculate \[-5 \times 8 = -40\] and \[5 \times (2y - 4) = 10y - 20.\]
This distribution clears the brackets, allowing for the further simplification of the expression.

Combining like terms

"Combining like terms" is crucial to consolidating expressions into simpler, more manageable forms. Like terms are terms that have the same variable raised to the same power.
Think of them like apples and oranges; you can combine apples with apples, and oranges with oranges, but not apples with oranges.
In the exercise, after using the distributive property, the expression \[6 - 40 + 10y - 20\] has like terms that need combining.
This involves:

• Add or subtract the constants: \[6 - 40 - 20 = -54\]
• The term \[10y\] remains as it has no other like terms.

Finally, you rewrite the expression by placing the term with the variable first, giving you \[10y - 54\].
Through these simple steps, combining like terms helps clean up and present the expression in its simplest form.