Problem 50 - Prealgebra

Question

Use the distributive property to combine similar terms. \(-3 x+9 x\)

Step-by-Step Solution

Answer

The expression simplifies to \(6x\).

Step 1: Identify Similar Terms

The problem involves the expression \(-3x + 9x\). Notice both terms have the same variable \(x\), which makes them like terms. We can combine these terms since they have the same variable factor.

Step 2: Distribute the Coefficient

The expression \(-3x + 9x\) can be combined by adding the coefficients. This means we group the coefficients \(-3\) and \(9\) together and keep the variable \(x\) as a factor. Essentially we are distributing \(x\), so the expression becomes \((-3+9)x\).

Step 3: Simplify the Coefficients

Perform the arithmetic operation on the coefficients: \(-3 + 9 = 6\). Now replacing in the expression, we have \(6x\). This simplification reduces the expression to \(6x\).

Key Concepts

Combining Like Terms

When dealing with algebraic expressions, you will often encounter the concept of "like terms." Like terms are terms that have the same variable raised to the same power. This means they share the same letters, and these letters have the same exponents.
In the expression \(-3x + 9x\), both terms are like terms because they both contain the variable \(x\) to the same power of 1. This commonality allows them to be combined into a single term.

• When combining, you only add or subtract the coefficients, while the variable part remains unchanged.
• This process helps in simplifying expressions and making them easier to work with.

Understanding how to combine like terms effectively is crucial for simplifying expressions and solving equations.

Coefficients

Coefficients are the numerical parts of the terms in an algebraic expression. In our example with the expression \(-3x + 9x\), the numbers \(-3\) and \(9\) are the coefficients of \(x\).
Coefficients tell you how many times to multiply the variable they are attached to.

• Consider \(-3x\): Here, \(-3\) indicates multiplying \(x\) by \(-3\).
• In \(9x\), the \(9\) tells us that \(x\) is multiplied by \(9\).
• Combining them involves adding these coefficients together: \(-3 + 9 = 6\).

Thus, when you combine, you are primarily working with the coefficients, resulting in the simplified expression \(6x\). Understanding coefficients aids in quickly combining like terms.

Simplifying Expressions

Simplifying an expression means reducing it to its most basic form without changing its value. It often involves a few key processes: combining like terms and performing arithmetic operations.
In the example \(-3x + 9x\), simplification allows us to transform these two terms into a simpler term:

• Firstly, notice the like terms (\(-3x\) and \(9x\)) and identify their coefficients.
• Then, perform the arithmetic operation to combine the coefficients: \(-3 + 9 = 6\).
• Finally, attach the variable \(x\) back to this new coefficient, resulting in \(6x\).

The simplified expression \(6x\) represents the original expression in a more compact form. Simplifying expressions is a foundational skill in algebra that makes equations easier to solve and understand. Always aim to simplify expressions as much as possible in any algebraic manipulation.