Problem 33 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$-(3 x-1)-2(5 x-1)+4(-2 x-3)$$

Step-by-Step Solution

Answer

The simplified expression is \(-21x - 9\).

Step 1: Distribute Negative Signs and Numbers

Start by distributing the negative signs and the numbers through the parentheses for each term. This means multiplying each term inside the parentheses by the number or negative sign outside. 1. \[-(3x-1) = -3x + 1\] 2. \[-2(5x-1) = -10x + 2\]3. \[4(-2x-3) = -8x - 12\]

Step 2: Combine Like Terms

Once you have distributed all terms, the expression now looks like this: \[-3x + 1 - 10x + 2 - 8x - 12\]Combine the like terms:For the \(x\) terms:\[-3x - 10x - 8x = -21x\]For the constant terms:\[1 + 2 - 12 = -9\]

Step 3: Write the Simplified Expression

Combine the results from Step 2 to write the simplified form of the expression:\[-21x - 9\]

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions means reducing the expression to its simplest form. This involves combining all like terms and performing any arithmetic operations. By simplifying, you make complex-looking algebraic expressions easier to understand and work with.

Consider the given expression:

• Start by addressing any parentheses or brackets to reveal all terms.
• Perform operations like distributing numbers or signs as needed.
• Look for like terms to combine and simplify the expression further.

The goal is a clear and concise expression with all similar terms combined, leaving no repeated terms.

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows you to multiply a single term and two or more terms inside a set of parentheses. Here's how it works:

The formula is simple: for any numbers or variables, \[a(b + c) = ab + ac\].

In the given exercise, you apply this property by multiplying the number outside the parentheses with each term inside.

• For \(- (3x - 1)\), multiply \(-1\) with each term: \(-3x + 1\).
• For \(-2(5x - 1)\), multiply \(-2\) with both terms: \(-10x + 2\).
• Lastly, for \(4(-2x - 3)\), multiply \(4\) with each: \(-8x - 12\).

The Distributive Property is key when simplifying algebraic expressions as it helps to eliminate parentheses.

Like Terms

In algebra, like terms refer to terms that have the same variable components raised to the same power. Simplifying means merging all these duplicate variables or numbers.

Look at these steps:

• Identify all the terms with the same variable component. In this case, terms with \(x\) are like terms: \(-3x, -10x, -8x\).
• Combine the coefficients of these terms to make one simplified term: \(-3x - 10x - 8x = -21x\).
• Don’t forget the constant terms. Here, combine \(1, 2, -12\) to get \(-9\).

By simplifying like terms in this manner, the expression becomes more manageable as you finalize the calculation.

Negative Signs

Dealing with negative signs can be tricky, but understanding their effect is crucial for simplification. A negative sign in algebra is important for changing the sign of a term, especially when outside brackets or a negative number.

Here's how you handle them:

• A negative sign by itself (e.g., \(- (3x - 1)\)) will reverse the signs of terms inside the brackets, changing \(3x - 1\) to \(-3x + 1\).
• When there’s multiplication with negatives, use them like regular numbers, adjusting the operation sign accordingly.
• Continuing through the entire expression, apply any negative operations to ensure accurate simplification.

Carefully tracking and applying negative signs is key to maintaining the integrity of an algebraic expression as you simplify it.