Problem 40 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 8 y-2-3(y+4) $$

Step-by-Step Solution

Answer

The simplified expression is \(5y - 14\).

Step 1 - Distribute Negative Sign

The expression given is \(8y - 2 - 3(y + 4)\). The first step to simplify the expression is to distribute the \(-3\) through the terms inside the parentheses. Remember that when you distribute a negative sign, it changes the sign of each term inside the parentheses. Hence, distribute \(-3\) to both \(y\) and \(4\): \(-3 \times y = -3y\)\(-3 \times 4 = -12\)Thus, the expression becomes \(8y - 2 - 3y - 12\).

Step 2 - Combine Like Terms

Now, we need to combine the like terms present in the expression \(8y - 2 - 3y - 12\). The like terms here are the terms containing \(y\) and the constant terms. Combine them:For the \(y\) terms: \( 8y - 3y = 5y \)For the constant terms:\( -2 - 12 = -14 \)This simplification leads to the expression: \( 5y - 14 \).

Key Concepts

Distributive Property

When you encounter a math problem that requires simplifying expressions, one powerful tool you'll use is the distributive property. This property helps to break down expressions that have parentheses, making them easier to work with. Essentially, when you see a number or variable outside a set of parentheses, you multiply each term inside the parentheses by that number or variable. For example, in the expression \(-3(y + 4)\), you distribute (or "multiply out") the \(-3\) through both terms inside the parentheses: \(-3 \times y = -3y\) and \(-3 \times 4 = -12\).

This changes the expression from \(8y - 2 - 3(y + 4)\) to \(8y - 2 - 3y - 12\). When you distribute, remember to pay attention to signs; a negative sign will switch the signs of all terms inside the parentheses.

Combining Like Terms

Once you've used the distributive property, the next step in simplifying an expression is to combine like terms. Like terms are terms whose variables and exponents (if they have them) are exactly the same. Only these terms can be combined through addition or subtraction.

In our example, the terms \(8y\) and \(-3y\) both include the variable \(y\) and can be combined. Similarly, the constant terms \(-2\) and \(-12\) are also like terms. By combining 8\(y\) and \(-3y\), you get \(5y\). By adding \(-2\) and \(-12\), you arrive at \(-14\). Each step simplifies the expression, leading to \(5y - 14\). Keep in mind that combining like terms is crucial for simplifying any algebraic expression.

Simplifying Expressions

Simplifying expressions is the process of making an equation or leftover expression as compact and efficient as possible. The goal is to reduce the complexity of the expression while ensuring it expresses the same value or relationship. Simplifying involves using various algebraic techniques, such as the distributive property and combining like terms, as we've seen.

Starting with our example, using the distributive property helps remove parentheses and set the stage for combining like terms, which further consolidates the expression. By reducing the expression to its simplest form \(5y - 14\), you've made further calculations, if needed, quicker and easier.

• First, work through any parentheses using distributive property or others.
• Second, combine any like terms to simplify further.

Remember, simplifying expressions reduces the potential for errors down the line when solving equations or inequalities.