Problem 47 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ -4(1-2 m+n)+4 $$

Step-by-Step Solution

Answer

The expression simplifies to \(8m - 4n\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). Here, you have \(-4(1 - 2m + n)\). Distribute \(-4\) to each term inside the parentheses: \(-4 \cdot 1\), \(-4 \cdot (-2m)\), and \(-4 \cdot n\).

Step 2: Multiply Each Term

Now, perform the multiplication for each distributed term: \(-4 \cdot 1 = -4\), \(-4 \cdot (-2m) = 8m\), and \(-4 \cdot n = -4n\). This gives you the expression: \(-4 + 8m - 4n\).

Step 3: Add the Constant

Notice the original expression also has a \(+4\) outside the parentheses. Add this \(+4\) to the expression obtained from the distributive property: \(-4 + 8m - 4n + 4\).

Step 4: Simplify the Expression

Combine like terms in the expression. The constants \(-4\) and \(+4\) combine to \(0\). The expression simplifies to: \(8m - 4n\).

Key Concepts

Simplifying Expressions

Simplifying expressions means making them as concise and straightforward as possible, while keeping their value unchanged. When simplifying, you'll often use mathematical properties and combine like terms. In our given exercise, after using the distributive property, we have the expression

• -4 + 8m - 4n + 4

To simplify this expression, first look for terms that are alike. Here, the numbers

• -4 and +4

These are constant terms that can be combined. When added, they equal zero. Since they cancel each other out, they no longer appear in the simplified expression. What remains of the expression is

• 8m - 4n

Remember, the aim of simplifying is to make the expression as simple and clear as possible, while maintaining its original value.

Like Terms

Like terms are terms that have the same variable(s) raised to the same power. They are combined during the simplification process to make expressions simpler. In the expression

• 8m - 4n

we have two terms, but notice they are not like terms. This is because

• 8m
• -4n

contain different variables, making them unlike terms. Like terms in algebra can usually be combined by adding or subtracting the coefficients, which are the numbers in front of the variables. For example,

• 3x and 4x

are like terms and can be combined to make

• 7x

When simplifying expressions, identifying and combining like terms help in reducing the expression's complexity and solving equations.

Variables in Algebra

Variables are symbols used in algebra to represent numbers that can vary within a set of possible values. They are usually denoted by letters such as

• m
• n

In our example, the expression

• 8m - 4n

includes variables, which means it cannot be reduced to a single numerical value without further information about the variables. Variables allow expressions to be general and flexible, representing real-world situations where a relationship between quantities exists. For instance, if you know the specific values for

• m and n

you can substitute these in to find the numerical result of the expression. Understanding variables is crucial in algebra, as they serve as placeholders for information you might discover or solve for through equations and problem-solving.