Problem 32 - Elementary Algebra

Question

Subtract \(x-3\) from twice the quantity \(x-1\).

Step-by-Step Solution

Answer

The result is \(x + 1\).

Step 1: Identify the Expressions

First, identify the expressions involved in the problem. We need to subtract \(x - 3\) from twice the quantity \(x - 1\). The expressions are:1. \(x - 3\)2. Two times \(x - 1\), which is \(2(x - 1)\).

Step 2: Simplify Twice the Quantity

Calculate twice the quantity \(x - 1\) by applying the distributive property:\[2(x - 1) = 2 \cdot x - 2 \cdot 1 = 2x - 2\]

Step 3: Set up the Subtraction

According to the problem, subtract \(x - 3\) from \(2x - 2\). This operation is represented as:\[2x - 2 - (x - 3)\]

Step 4: Distribute and Simplify

Apply the distribution of the negative sign in the subtraction:\[2x - 2 - x + 3\]Now, combine like terms:\[ (2x - x) + (-2 + 3) = x + 1 \]

Step 5: Final Result:

After combining like terms, the final simplified expression is:\[x + 1\]

Key Concepts

Distributive Property

The distributive property is a fundamental rule used in algebra to simplify expressions. It's most commonly represented as \(a(b + c) = ab + ac\), which means you distribute or "multiply out" the factor \(a\) to both \(b\) and \(c\). This property is incredibly useful for transforming expressions into simpler forms.

In the given exercise, we apply the distributive property to find "twice the quantity \(x-1\)". This means multiplying 2 by each term inside the parentheses. The calculation follows this pattern:

• Double the \(x\) term: 2 multiplied by \(x\), giving us \(2x\).
• Double the constant: 2 multiplied by -1, which gives us -2.

Putting these together, the expression \(2(x - 1)\) transforms into \(2x - 2\). This simplifies our task of subtraction later on.

Simplifying Expressions

Simplifying expressions involves breaking down complex algebraic expressions into their simplest form. This often requires using properties like the distributive property and combining like terms. Simplification helps in making calculations easier to manage.

For the problem at hand, once we found \(2(x - 1)\) as \(2x - 2\), we maintain clarity by ensuring all expressions are in their simplest form before proceeding with any operations. Simplifying means:

• Performing basic arithmetic to resolve any factors.
• Ensuring terms are organized for further operations like subtraction or addition.

This step ensures the groundwork is laid for easy manipulation in subsequent operations, such as subtraction.

Subtraction of Expressions

Subtracting one expression from another involves careful consideration of each term. In algebra, subtraction is not just about "taking away" numbers; it often involves distributing a negative sign through an expression.

The problem specifically asks us to subtract \(x - 3\) from \(2x - 2\). This process is represented as subtracting each term in \(x - 3\) from \(2x - 2\), requiring careful attention to signs:

• Distribute the negative sign across \(x - 3\), turning it into \(-x + 3\).
• Combine with the initial expression: \(2x - 2 - x + 3\).

Keeping track of positive and negative signs during subtraction is crucial to avoid errors, especially when similar terms need to be combined.

Combining Like Terms

Combining like terms is an essential skill for simplifying expressions further once operations like addition or subtraction are performed. Like terms have identical variable parts, meaning they can be added or subtracted easily.

After distributing the negative sign in the subtraction step, the expression becomes \(2x - x + 3 - 2\). Here, combining like terms involves:

• Adding and subtracting terms with \(x\): Combine \(2x - x\), resulting in \(x\).
• Managing constant terms: Combine -2 and +3, resulting in +1.

The final expression, \(x + 1\), is the simplest form of the original problem, showcasing how combining like terms streamlines algebraic expressions for clarity and ease of further mathematical operations.