Problem 85 - Algebra A Combined Function

Question

Perform the indicated operations. \((3 x-1)-(10 x-6)\)

Step-by-Step Solution

Answer

The result is \(-7x + 5\).

Step 1: Distribute the Negative Sign

First, distribute the negative sign across the terms in the second parenthesis. This changes the expression from \[(3x - 1) - (10x - 6)\] to \[(3x - 1) - 10x + 6\].

Step 2: Combine Like Terms

Next, combine the like terms. The like terms are the terms with the variable \(x\) and the constant terms:- Combine \(3x\) and \(-10x\): \(3x - 10x = -7x\).- Combine \(-1\) and \(+6\): \(-1 + 6 = 5\).Now the expression simplifies to \[-7x + 5\].

Key Concepts

Distributing Negative Signs

When working with algebraic expressions, one crucial step is properly distributing negative signs. A negative sign in front of a parenthesis affects every term within those parentheses. To understand this better, let's break it down:

• When you see a negative sign such as \((a - (b + c))\), imagine there is a "hidden" -1 being multiplied to everything inside the parenthesis.
• Apply the distributive property, which means multiplying each term inside the bracket by -1. For example, \(-(b + c)\) becomes \(-b - c\).

So in the example \((3x - 1) - (10x - 6)\), distributing the negative sign turns it into \((3x - 1) - 10x + 6\).
This is because \(-(10x - 6)\) becomes \(-10x + 6\). Mastering this step is essential for simplifying algebraic expressions accurately.

Combining Like Terms

Once negative signs have been correctly distributed, the next step is to combine like terms. But what are like terms? Like terms are terms in an algebraic expression that have the same variables raised to the same power.

• For example, \(3x\) and \(-10x\) are like terms because they both have the variable \(x\).
• Similarly, constant terms like \(-1\) and \(+6\) are also considered like terms because neither of them have a variable attached.

To combine like terms, simply add or subtract their coefficients. In our expression \(3x - 1 - 10x + 6\), you combine:

• \(3x\) and \(-10x\) to get \(-7x\).
• \(-1\) and \(+6\) to get \(+5\).

This reduces the expression to \(-7x + 5\). Being able to identify and combine like terms is key to simplifying and solving algebraic expressions.

Simplifying Expressions

Simplifying expressions involves taking a potentially complex algebraic statement and rewriting it in the simplest form. This process includes distributing negative signs, combining like terms, and sometimes performing basic arithmetic.
Simplification is essential for understanding and solving mathematical problems, making them more manageable. Here's how it helps:

• Reduces clutter: An expression like \(3x - 10x - 1 + 6\) becomes \(-7x + 5\) — much easier to interpret at a glance.
• Aids in solving equations: Simplified equations are easier to manipulate when isolating variables or finding solutions.

In our example, by first addressing the parentheses and then combining like terms, we efficiently pared down the expression. The result \(-7x + 5\) represents the same relationship but is far more concise. Mastery of simplification strengthens overall problem-solving skills in algebra.