Problem 363 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ -(3 x-7) $$

Step-by-Step Solution

Answer

-3x + 7

Step 1: Understand the Distributive Property

The Distributive Property states that for any numbers a, b, and c, the expression \(a(b + c) = ab + ac\). This also works for subtraction: \(a(b - c) = ab - ac\).

Step 2: Distribute the Negative Sign

Here, we need to distribute the negative sign to both terms inside the parentheses. The expression is \(-(3x - 7)\). This means we will multiply -1 by both 3x and -7.

Step 3: Multiply Each Term Inside the Parentheses

Distribute \(-1\) to \(3x\): \(-1 \times 3x = -3x\) and to \(-7\): \(-1 \times -7 = 7\).

Step 4: Combine the Results

After distributing, combine the terms to get \(-3x + 7\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions can seem tricky at first, but it's just about breaking down and combining the parts of an equation. Consider the expression we need to simplify: e.g., \(-(3x-7)\)The primary goal here is to reduce it to its simplest form. We do this by using certain mathematical properties like the Distributive Property. This property helps in managing parentheses in algebra. When we simplify expressions, we typically do the following:

• Remove the parentheses by distributing the factors outside the parentheses to each term inside.
• Combine any like terms, which adds or subtracts similar variables.

By mastering these steps, you can make complex expressions easier to handle.

Negative Multiplication

Negative multiplication might sound a bit tough, but it follows simple rules. When you multiply a negative number by another number (positive or negative), it changes the sign:

• Negative times Positive = Negative (e.g., \(-1 \times 3x = -3x\)).
• Negative times Negative = Positive (e.g., \(-1 \times -7 = 7\)).

In the distributive property context, this means every term inside the parentheses will change sign if multiplied by -1. For the expression e.g., \(-(3x-7)\), when we distribute -1, -3x turns negative and -7 turns positive, giving us -3x + 7.

Combining Like Terms

Combining like terms is the key to simplifying algebraic expressions fully. Like terms are terms that have the same variables raised to the same power.For example:

• \(3x\) and \(4x\) are like terms.

When you simplify, you add or subtract the coefficients (numbers in front) of these like terms. In our expression e.g., \(-(3x-7)\), after distributing -1, we get -3x and 7. Since there are no other 'x' terms or constant terms to combine with -3x and 7 respectively, our simplified form is just \-3x + 7\.