Problem 27 - Elementary Algebra

Question

Multiply. $$ -2(3 x-2 x 2+x-3) $$

Step-by-Step Solution

Answer

The simplified expression is \(-4x + 6\).

Step 1: Distribute the Negative Sign Inside the Parentheses

First, we distribute \(-2\) to each term inside the parentheses. This preparation helps us calculate \[-2(3x) + (-2)(-2x) + (-2)(x) + (-2)(-3)\]. This property stems from the distributive law which states \(a(b+c)=ab+ac\).

Step 2: Perform the Sign Multiplications

Calculate the multiplication step-by-step: 1. \(-2 \times 3x = -6x\).2. \(-2 \times -2x = 4x\).3. \(-2 \times x = -2x\).4. \(-2 \times -3 = 6\).

Step 3: Combine Like Terms

Next, we combine the terms with x: \(-6x + 4x - 2x\) sums up to \(-4x\). Finally, add the constant: \(-4x + 6\).

Key Concepts

Understanding Multiplication in Distributive Property

Multiplication is a core arithmetic operation and a vital component when applying the distributive property. This property lets you multiply a single term across a sum or difference within parentheses, effectively distributing the multiplication to each term inside. For example, with

• -2(3x) = -6x
• -2(-2x) = 4x
• -2(x) = -2x
• -2(-3) = 6

Each equation results from multiplying -2 by each of the terms in the parentheses. Notice how we utilized multiplication on numbers and coefficients associated with variables. By doing so, we systematically break down complex expressions into simpler parts, achieving a clearer understanding for further calculations.
Remember, treating all terms individually while preserving their signs is crucial. This is especially needed when handling negative numbers or coefficients.

Identifying and Combining Like Terms

In algebra, 'like terms' are terms that have the same variable part. These terms can be combined using simple addition or subtraction. For instance, terms such as -3x, 5x, and 8x are like terms because they all have the common variable 'x'.
In expressions, once multiplication is completed using the distributive property, like terms are often combined to simplify the expression.
For example, after distributing:

• -6x
• 4x
• -2x

we can combine these into a single term: -6x + 4x - 2x = -4x
Unlike, numbers without variables, known as constants, cannot be combined with terms that include variables such as 'x'.
Always set aside constants, like the number 6 in this expression, until you finalize combining like terms; then, add or subtract it as necessary to complete the simplification.

Handling Negative Numbers in Calculations

Negative numbers introduce a layer of complexity in algebra, especially during multiplication and combination of terms.
In our exercise, multiple steps involved negative multiplication, such as multiplying -2 with terms inside the parentheses. The fundamental rule here is:

• A negative multiplied by a positive gives a negative result, e.g., -2 × 3x = -6x.
• A negative multiplied by a negative always gives a positive result, as seen: -2 × (-2x) = 4x.

Handling these appropriately ensures the correct simplification of expressions.
Keep an eye on your negative signs throughout the calculation process. Simple errors can propagate and cause the entire solution to go awry.
Therefore, consistently practice with the goal to enhance accuracy while dealing with negative numbers.