Problem 36 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$6(4 x-2)+7$$

Step-by-Step Solution

Answer

The simplified expression is \(24x - 5\).

Step 1: Distribute the 6 into the Parentheses

The exercise involves distributing the number outside the parentheses, 6, to each term inside the parentheses. This means multiplying 6 by both terms inside the parentheses: \[6(4x) = 6 \times 4x = 24x\]\[6(-2) = 6 \times -2 = -12\]

Step 2: Combine the Distributed Terms

After distributing, you replace the original expression with the distributed terms. The expression becomes: \[24x - 12 + 7\]

Step 3: Simplify the Expression by Combining Like Terms

Combine the constant terms in the expression. In this case, that means combining \(-12\) and \(+7\). \[-12 + 7 = -5\] So the simplified expression is \[24x - 5\].

Key Concepts

Combining Like Terms

In algebra, combining like terms is a crucial step in simplifying expressions. To "combine like terms" means grouping and simplifying terms in an expression that have the same variable raised to the same power. For example, in the expression \(24x - 12 + 7\), once you distribute the numbers, the like terms are the constants \(-12\) and \(+7\). By combining these two, you can simplify the expression, reducing

• Complexity
• potential errors in further calculations
• the number of terms to handle

To combine \(-12\) and \(+7\), simply add them together, resulting in \(-5\). This makes your expression neater and easier to work with. Always remember to only combine terms that have the same characteristics; otherwise, you will end up with incorrect results.
In short, combining like terms helps in organizing an expression, laying a strong foundation for solving algebraic equations.

Simplifying Algebraic Expressions

Simplifying an algebraic expression is all about making the expression as straightforward and clear as possible. It involves a couple of steps that you typically repeat in any problem. As demonstrated in the exercise \(6(4x-2)+7\), after distributing, you're left with a new expression: \(24x - 12 + 7\).
The next major step is to simplify this further. By combining like terms, the distracting numerical clutter disappears, and you are left with fewer terms: \(24x - 5\).
When expressions are simplified, they become more manageable. You can think of it as tidying your room: less clutter means you can find things faster and without stress. It reduces the paper complexity you have to deal with and makes subsequent algebraic manipulations easier to perform.

• Gathering and adding like terms
• Ensuring each term is as simple as it can be
• Finalizing the expression in its simplest form

Mastering simplification is essential in algebra, as it makes solving equations much more efficient. As an algebra student, practicing simplification strengthens your overall math skills, providing a solid base for future problem-solving.

Basic Algebra Concepts

Getting comfortable with basic algebra concepts sets the stage for tackling more complex problems. One of the foundational blocks is understanding how numbers and variables interact within expressions. Just as in our example, it's crucial to properly apply arithmetic rules like the distributive property.
The distributive property is one of the primary tools. It allows you to multiply a single term by several terms inside parentheses, distributing the multiplication across each of them. In our example, using the distributive property, \(6\) is multiplied separately by \(4x\) and \(-2\), resulting in \(24x - 12\).
Breaking down expressions, identifying terms to combine, and reassembling them are everyday tasks in algebra.

• Recognizing terms that are "like terms"
• Applying arithmetic properties correctly
• Practicing routine simplification steps

This preparation is vital whether you're solving simple linear equations or preparing for more advanced calculus operations. Master these basics, and you'll have a head start in understanding the language of mathematics.