Problem 71 - Elementary Algebra

Question

Simplify. $$ 2 x-(3 x-4 y-1) $$

Step-by-Step Solution

Answer

\(-x + 4y + 1\)

Step 1: Distribute the Negative

First, distribute the negative sign inside the parentheses across each term. This involves changing the sign of each term inside the parentheses. The expression becomes:\[ 2x - 3x + 4y + 1 \]

Step 2: Combine Like Terms

Next, identify and combine like terms in the expression. The expression contains like terms "\(2x\)" and "\(-3x\)". Combine these terms:\[ 2x - 3x = -x \]So the expression simplifies to:\[ -x + 4y + 1 \]

Key Concepts

Simplification

Simplification in algebra is about making an expression shorter and easier to work with without changing its value. This process helps to see what's happening in the expression more clearly. To simplify a mathematical expression means to break down and combine its components to reach a form that can easily be interpreted and manipulated.

Here are the key steps involved:

• Breaking down complex expressions into smaller, more manageable parts.
• Combining like terms to streamline the expression.
• Eliminating unnecessary parentheses, using the distributive property when needed.

By applying these steps, you are able to see the essential components of the expression and understand it better. Simplification is a foundational skill in algebra that helps students to easily solve equations and inequalities.

Distributive Property

The distributive property is a key concept in algebra that allows you to simplify expressions by distributing a value across terms inside parentheses. When you see an expression like the one in our exercise, the property helps manage terms more efficiently.

This property states that a term outside the parentheses can be multiplied by each term inside the parentheses:\[a(b + c) = ab + ac\]In the exercise:

• The negative sign in front of the parentheses \( - (3x - 4y - 1) \) needs to be distributed.
• This results in changing the signs of each term, converting to \( -3x + 4y + 1 \).

Using this property makes the next steps, like combining like terms, much simpler. It’s particularly useful in algebra when dealing with expressions with multiple terms placed together by parentheses.

Like Terms

Like terms are components of an algebraic expression that have the same variable factors. Combining like terms is a crucial part of simplifying algebraic expressions. By lumping similar terms together, you create a simpler expression that's easier to solve or analyze.

Here’s a quick way to understand it:

• "Like terms" share the same variables raised to the same powers. For example, \( 2x \) and \( -3x \) are like terms because they both contain the variable \( x \).
• Constant terms, such as numbers without variables, are also like terms. For instance, \( +1 \) is a constant term in the exercise.

To combine, simply perform the basic operations on the coefficients of these terms. In the exercise: \[2x - 3x = -x\]Combining like terms simplifies the expression and makes the equation more manageable. It helps in reducing complex statements to forms that are straightforward to interpret.