Problem 13 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (12 x-7)-(5 x-12) $$

Step-by-Step Solution

Answer

The simplified expression is \(7x + 5\).

Step 1: Write Down the Expression

The expression we need to operate on is \[(12x - 7) - (5x - 12).\]

Step 2: Distribute the Negative Sign

Distribute the negative sign across the second parentheses. This changes the sign of each term inside the parentheses, transforming the expression to:\[12x - 7 - 5x + 12.\]

Step 3: Combine Like Terms

Now, we combine the like terms. Start by combining the terms with \(x\):\[12x - 5x = 7x.\]Then combine the constant terms:\[-7 + 12 = 5.\]

Step 4: Write the Simplified Expression

Combine the results from the previous step to form the final simplified expression:\[7x + 5.\]

Key Concepts

Distributive Property

The distributive property is a useful algebraic tool that helps when dealing with expressions involving parentheses. This property allows you to distribute a factor across the terms inside parentheses by multiplying it with each term. In the exercise, the negative sign in front of the second set of parentheses must be distributed to each term inside the parentheses. This transforms

• \((5x)\) into \(-5x\)
• \((-12)\) into \(+12\)

With the negative sign distributed, the expression becomes easier to manage. It is now transformed into \(12x - 7 - 5x + 12\). Understanding the distributive property is essential for simplifying various algebraic expressions.

Like Terms

"Like terms" in algebra are terms that have the same variables raised to the same power. Only like terms can be combined or simplified. In the given exercise, we identify "like terms" to simplify the expression into an easier form.

• \(12x\) and \(-5x\) are like terms because they both contain the variable \(x\).
• \(-7\) and \(+12\) are like terms because they are both constant numbers with no variables attached.

After identifying these terms, you can combine them: \(12x - 5x = 7x\) for the variable terms, and \(-7 + 12 = 5\) for the constants. Combining like terms is a fundamental step in algebra that helps to simplify expressions.

Simplification

Simplifying an algebraic expression means making it as concise as possible. This involves combining like terms, distributing factors, or removing unnecessary elements from expressions. In the provided exercise, simplification boils down to a few key steps:

• Distributing the negative sign through the parentheses.
• Combining the like terms: both the \(x\)-terms and the constant numbers.
• Writing the final, simplified expression: \(7x + 5\).

Simplification makes dealing with the expression easier, and it is a foundational skill required when solving math problems. Simplifying correctly can often make complex problems far more manageable.

Polynomial Operations

Polynomial operations involve adding, subtracting, multiplying, or dividing polynomials. In this exercise, we focus on subtracting one polynomial from another. Here's how it works:First, write down the expression, which consists of two polynomials. In our case, it is \(12x - 7\) minus \(5x - 12\). Subtraction involves changing the signs of the second polynomial before combining it with the first. Here, distribute the negative sign: transforming \((5x - 12)\) into \(-5x + 12\). Then, combine the resultant like terms, leading to \(7x + 5\).These steps illustrate a basic form of polynomial operation, which once understood, can apply to more complex algebraic problems. In practice, understanding how to maneuver these operations is crucial in mathematics.