Problem 66 - Algebra 1

Question

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ -4(y+2)-6 y $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-10y - 8\).

Step 1: Distribute the Multiplication

First, distribute \(-4\) across \(y + 2\), which breaks down into \(-4 * y - 4 * 2\). This simplifies to \(-4y - 8\). Now the original equation becomes \(-4y - 8 - 6y\).

Step 2: Combine Like Terms

Next, combine the like terms, which in this case are the variables \(-4y\) and \(-6y\), these simplify to \(-10y\). Thus the equation is now \(-10y - 8\).

Key Concepts

Distributive Property

The distributive property is a fundamental building block in algebra that helps us to minimize confusion when dealing with an expression involving a product and a sum or difference. Imagine you have \(a(b + c)\), the distributive property tells us that this can be rewritten as \(ab + ac\). This is a simple way to spread the multiplication across all terms within parentheses.

For example, consider the expression \(-4(y+2)\). By distributing \(-4\), you'll multiply each term inside the parentheses. So, it becomes:

• \(-4 * y = -4y\), and
• \(-4 * 2 = -8\)

The expression now simplifies to \(-4y - 8\). Your initial complex expression has become more manageable! This is how the distributive property helps streamline calculations and makes it easier to handle algebraic expressions.

Simplification

Simplification is the process of reducing equations or expressions into their simplest form. Once you've applied the distributive property, the next step is to simplify the expression further by combining like terms.

In the example given, we have the expression \(-4y - 8 - 6y\). To simplify this, we must combine any terms that are alike. Like terms are terms that have the same variables raised to the same power.

• Both \(-4y\) and \(-6y\) are like terms because they have the "y" variable.
• Their coefficients can be directly added or subtracted, so we combine them: \(-4y - 6y = -10y\).

Now your expression is simplified to \(-10y - 8\). Simplification is about making expressions as clean and concise as possible, making them easier to interpret and solve.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are the language of algebra, enabling us to describe mathematical ideas in a concise format.

Take the example expression: \(-4(y+2)-6y\). This expression includes:

• A constant number part: like "2" and "-8" in the final solution.
• Variable parts: like "y" which shows an unknown that can change values.

It is important to understand how these pieces interact.Algebraic expressions can be manipulated in various ways: using properties like distributive, combining like terms, and more. These operations help us solve for unknowns and reach desired outcomes in equations.Moreover, mastering algebraic expressions lays the groundwork for handling more complex mathematical tasks. It's akin to mastering vocabulary in a new language, allowing you to "speak" algebra fluently and solve a wider range of problems.