Problem 44 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ (8-5 y)-(4+3 y) $$

Step-by-Step Solution

Answer

The simplified expression is \(4 - 8y\).

Step 1: Distribute the Negative Sign

The given expression is \((8 - 5y) - (4 + 3y)\). First, we need to distribute the negative sign across the second set of parentheses. This changes the expression to \(8 - 5y - 4 - 3y\).

Step 2: Combine Like Terms

Now that parentheses have been removed, we'll combine like terms. Start with the constant terms: \(8 - 4 = 4\). Next, combine the \(y\)-terms: \(-5y - 3y = -8y\).

Step 3: Write the Final Simplified Expression

The simplified expression, combining all like terms, is \(4 - 8y\).

Key Concepts

Simplifying Expressions

When working with algebraic expressions, one of the primary goals is to simplify them. Simplifying expressions means to perform operations and combine all possible terms, reducing the expression to its simplest form. Consider the expression \[ (8 - 5y) - (4 + 3y) \]from the original exercise. We needed to simplify it by removing parentheses and combining terms. This process helps in making complicated expressions easier to work with or solve. Simplification involves:

• Removing parentheses by applying distribution, if necessary.
• Combining like terms to consolidate the expression.

Ultimately, the simplified expression should contain the fewest terms possible, making further algebraic manipulations much easier.

Like Terms

In algebra, like terms are terms that have identical variable parts. For example, in the expression \( 8 - 5y - 4 - 3y \),the terms \(-5y\)and \(-3y\)are like terms because they both involve the variable \(y\)raised to the same power (which is 1 in this case).Here's how you identify and combine like terms:

• Look for terms that have the same variable and power. For instance, \(-5y\)can be combined with \(-3y\)because both have \(y\).
• Constant terms, like \(8\)and \(-4\),are also considered like terms as they don’t have any variable.
• You can add or subtract like terms by simply combining their coefficients. So, \(-5y - 3y = -8y\)and \(8 - 4 = 4\).

Combining like terms is crucial in simplifying expressions as it reduces the number of terms to the simplest form.

Distributive Property

The distributive property in algebra is a useful tool that allows you to multiply a single term across terms within parentheses. In the context of the original expression,\[ (8 - 5y) - (4 + 3y) \],we apply this property indirectly by distributing the negative sign. Here's how the distributive property works:

• Multiply the term outside the parentheses by each term inside. If you have \(-1\) outside, multiply each inside term by \(-1\).
• For instance, applying in the expression gives us \(-(4 + 3y) = -4 - 3y\).The expression then transforms to \(8 - 5y - 4 - 3y\).
• This step ensures that all terms are correctly added or subtracted when combining later.

Understanding the distributive property is fundamental for managing expressions, especially when dealing with terms in parentheses. It allows you to extend the expression so you can proceed with combining like terms.