Problem 71 - Prealgebra

Question

Simplify each of the following expressions as much as possible. $$6(2 x-1)+4 x$$

Step-by-Step Solution

Answer

The simplified expression is \(16x - 6\).

Step 1: Distribute through the Parentheses

First, distribute the 6 across the terms inside the parentheses. This means you multiply 6 with each term inside: \[ 6 \cdot (2x - 1) = 6 \cdot 2x - 6 \cdot 1 \]This simplifies to:\[ 12x - 6 \]

Step 2: Combine Like Terms

Now, combine the like terms in the expression. The expression is currently:\[ 12x - 6 + 4x \]Combine the terms that involve \(x\):\[ (12x + 4x) - 6 \]This simplifies to:\[ 16x - 6 \]

Key Concepts

Distributive Property

When you see a number outside of parentheses and an expression inside, it is a signal to use the distributive property. This property says that you must multiply the term outside the parentheses by each term inside. In the exercise, we have the expression \(6(2x - 1)\).

To apply the distributive property here, multiply \(6\) by \(2x\) and then multiply \(6\) by \(-1\). This will give you:\[ 6 \cdot 2x - 6 \cdot 1 = 12x - 6 \]

The distributive property helps break down expressions into simpler parts and is a vital skill in algebra.

Combining Like Terms

Once you have used the distributive property, your next step is to look for like terms. Like terms are terms that contain the same variables raised to the same power. You can only combine these terms. In our current expression, \(12x - 6 + 4x\), notice the terms containing \(x\):\[ 12x \quad \text{and} \quad 4x \]

So, combine \(12x\) and \(4x\) by adding their coefficients (the numbers in front of the \(x\)) together:\[ 12x + 4x = 16x \]

Remember, terms that are constant numbers, like \(-6\), do not combine with terms that have variables. Combining like terms simplifies expressions to make them easier to work with.

Simplification of Expressions

Simplification is the process of reducing an expression to its most basic form. After distributing and combining like terms, we have the expression \(16x - 6\).

To check if any more simplification is possible, look for any other like terms or factors that can be further reduced. In this example, \(16x\) and \(-6\) are not like terms and cannot be combined further, so \(16x - 6\) is the simplified expression.

Simplifying expressions:

• Makes them easier to understand and work with.
• Helps in solving equations more efficiently.

Remember, the goal of simplification is to create the simplest, most readable form of an expression.