Problem 37 - Elementary Algebra

Question

Simplify. $$ 3 x+4 x-8-28-x $$

Step-by-Step Solution

Answer

The simplified expression is \(6x - 36\).

Step 1: Identify Like Terms

In the expression \(3x + 4x - 8 - 28 - x\), identify like terms. The like terms are coefficients of \(x\), and constants.\(3x\), \(4x\), and \(-x\) are like terms involving \(x\), and \(-8\) and \(-28\) are constant terms.

Step 2: Combine Like Terms with \(x\)

Add the coefficients of the like terms that include \(x\).\(3x + 4x - x = (3 + 4 - 1)x = 6x\).

Step 3: Combine Constant Terms

Add the constant terms:\(-8 - 28 = -36\).

Step 4: Write the Simplified Expression

The simplified expression from combining the terms is \(6x - 36\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing complexity to make them easier to work with. The goal is to express the algebraic equation in its simplest form without changing its value. This means we want a form that has fewer terms, less clutter, and is easy to interpret.

When simplifying, we focus on:

• Identifying and combining like terms
• Simplifying coefficients
• Reducing constants

The end result is a more organized and efficient algebraic expression, ready for further use. Think of it as cleaning up a messy room; you're organizing similar items together to create a neat and tidy space!

Like Terms

In algebra, terms are considered 'like' when they have the same variables raised to the same power. This means that the variables and their exponents are identical, allowing these terms to be added or subtracted. For example:

• Terms like \(3x\), \(5x\), and \(-x\) are considered like terms because they all contain the variable \(x\).
• On the other hand, \(2x\) and \(2y\) are not like terms because they involve different variables.

Identifying like terms is a crucial step in simplifying algebraic expressions, as it allows you to combine them into a single term. Imagine gathering all your apples and putting them in one basket; that's like combining your like terms!

Combining Like Terms

Once you identify like terms, combining them is straightforward. You simply add or subtract their coefficients while keeping the variable part unchanged. This process helps to condense the expression:

• For example, in the expression \(3x + 4x - x\), you combine the coefficients of the like terms to get \(3 + 4 - 1)\) which simplifies to \(6x\).
• For constant terms, like \(-8 - 28\), you simply perform the arithmetic to get a new combined constant, \(-36\).

By combining like terms, you streamline the expression into a much simpler form which is easier to handle. Think of it as merging several similar items into one, enhancing the efficiency of your expression!

Algebraic Coefficients

Algebraic coefficients are the numerical part of the terms in an expression. They indicate how many times a variable is multiplied by that number. In the expression \(3x + 4x - x\):

• The coefficients are \(3\), \(4\), and \(-1\) (the unseen coefficient of \(-x\) is \(-1\)).
• Unlike constant terms, coefficients are attached to variables and guide you in the process of combining terms.

Understanding coefficients is essential because they play a significant role in determining how like terms are combined. By focusing on the coefficients, you ensure that your simplification process yields accurate results. It's like knowing the exact number of groups of items you have before you join them together into a pile.