Problem 46 - Elementary Algebra

Question

Simplify. $$ x x 2-2 x-3+2 x-3 $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x - 6 \).

Step 1: Distribute the Terms

Start by distributing the terms in the expression. The given expression is \( x \times 2 - 2x - 3 + 2x - 3 \). Distribute \( x \times 2 \) to simplify this part of the expression to \( 2x \).

Step 2: Combine Like Terms

Now the expression reads \( 2x - 2x - 3 + 2x - 3 \). Begin combining like terms. Notice \( 2x \) and \(-2x\) will cancel each other, and \( -3 + 2x - 3 \) can be further simplified.

Step 3: Simplify Remaining Terms

Combine the constant terms \(-3 - 3\) to get \(-6\). What remains from the expression is \( 2x \).

Step 4: Final Simplified Expression

After simplification, the expression reduces to \( 2x - 6 \).

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that helps to break down expressions where multiplication and addition co-exist. It allows you to simplify expressions by distributing a factor across terms inside parentheses or being multiplied by a variable.
Let’s put this into simpler terms. With the example of:

• Distributing means spreading the multiplier over the terms inside the parentheses.
• The typical form is expressed as: \( a(b + c) = ab + ac \).
• This showcases how multiplication distributes over addition.

In the exercise you saw, we distributed the multiplication: \( x \times 2 \) to become \( 2x \). This allows us to simplify the expression further. By recognizing and using the distributive property, you can effectively manage how you handle and simplify algebraic expressions.

Combining Like Terms

Combining like terms is an essential algebraic technique that involves grouping similar terms to simplify an expression. Like terms are terms that have the exact variable part raised to the same power. Here’s a simple way to look at it:

• "Like terms" share the same variable and exponent but may have different coefficients.
• It’s like adding apples with apples; you can’t add apples to bananas because they’re not alike.

In an expression such as \( 2x - 2x - 3 + 2x - 3 \), you see like terms such as \( 2x \), \( -2x \), and another \( +2x \). By combining them, we simplify the algebraic expression. Notice:

• Positive and negative terms of the same variables, like \( 2x \) and \(-2x\), cancel each other out.
• What remains is the expression with fewer terms.

By combining like terms effectively, you simplify the expression while still maintaining its equality.

Constants in Algebra

Constants in algebra refer to numbers on their own without any variables attached to them. They remain unchanged no matter what the values of the variables in the expression are. Understanding constants are crucial because:

• They represent fixed values.
• Every algebraic expression will likely have at least one constant.

In our example, we dealt with the constants \(-3\) and another \(-3\), which is easily simplified by adding them together to give \(-6\). Handling constants involves straightforward arithmetic as these numbers do not depend on the variable in any sense.Recognizing and working with constants allows a cleaner and more manageable expression. They serve as the numerical backbone of any algebraic equation or expression, being readily combined and reduced where possible.