Problem 68 - Elementary Algebra

Question

Simplify. $$ -2(4 y+2)-3 y $$

Step-by-Step Solution

Answer

The simplified expression is \\(-11y - 4\\).

Step 1: Distribute the Constant

Apply the distributive property to remove the parentheses. Multiply \(-2\) by each term inside the parentheses: \(-2(4y + 2) = -2 \cdot 4y + (-2) \cdot 2\). This simplifies to \(-8y - 4\).

Step 2: Combine like terms

Combine all like terms from the resulting expression. The expression now is \-8y - 4 - 3y\. Combine the \(-8y\) and \(-3y\) terms: \(-8y - 3y = -11y\). Write the final expression: \(-11y - 4\).

Key Concepts

Understanding the Distributive Property

The distributive property is a useful tool in algebra that helps us to simplify expressions by removing parentheses. It states that for any numbers or variables \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). This means you multiply the term outside the parentheses by each term inside.
In our exercise, we have \(-2(4y + 2)\). Using the distributive property, we multiply \(-2\) by \(4y\) to get \(-8y\), and \(-2\) by 2 to get \(-4\).
This eliminates the parentheses, leaving us with \(-8y - 4\).

• Remember: Always apply the sign in front of the number. Here, \(-2\) is negative, so it affects the signs of the terms inside the parentheses.
• Think of it as spreading \(-2\) across both terms inside the parentheses.

Understanding this property will make it much easier to work with longer expressions.

Identifying and Combining Like Terms

Combining like terms is an important part of simplifying algebraic expressions. Like terms are terms in an expression that have the same variable raised to the same power. For example, \(-8y\) and \(-3y\) are like terms because both have the variable \(y\) raised to the first power.
In the expression \(-8y - 4 - 3y\), after distributing, we combine the like terms \(-8y\) and \(-3y\).
This is done by adding their coefficients: \(-8y - 3y = -11y\).

• Look for terms with the same variable and power.
• Only the coefficients (the numbers in front of the variables) are added or subtracted.

By combining like terms, the expression becomes simpler and easier to understand.

Simplifying Expressions: Bringing It All Together

Simplifying expressions is the process of combining and reducing expressions to their simplest form. By using properties like distribution and combining like terms, we make complex expressions more manageable.
In our example, we started with \(-2(4y + 2) - 3y\).
First, we applied the distributive property to get \(-8y - 4\). Then, we combined the like terms \(-8y\) and \(-3y\) to end up with the simplified expression \(-11y - 4\).

• Simplifying helps in solving equations more easily.
• Always follow a step-by-step approach: Distribute, combine, simplify.
• This process reduces errors and increases understanding.

Simplifying expressions is a key skill in algebra, making calculations and problem-solving more straightforward.