Problem 81 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(13x - 7\).

Step 1: Identify Like Terms

In the expression \((3x - 1) + (10x - 6)\), notice that \(3x\) and \(10x\) are like terms (both contain \(x\)). Similarly, \(-1\) and \(-6\) are constant terms.

Step 2: Combine Like Terms

Add the like terms together: \(3x + 10x\) and \(-1 - 6\). Calculate each sum separately. For the \(x\) terms: \(3x + 10x = 13x\). For the constants: \(-1 - 6 = -7\).

Step 3: Write the Simplified Expression

Combine the results of the additions to form the final simplified expression: \(13x - 7\).

Key Concepts

Like Terms

In algebra, like terms are terms that have the same variable raised to the same power. Identifying like terms is crucial because it allows us to simplify expressions by combining them. For instance, in the expression \((3x - 1) + (10x - 6)\), the terms \(3x\) and \(10x\) are considered like terms since they both have the variable \(x\) with the same exponent (which is one in this case). The terms \(-1\) and \(-6\) are like terms too, but they are constants (numbers without variables).

By focusing on the variables and their exponents, you can quickly determine which terms in an expression are alike. Remember that coefficients, the numbers before the variables, do not need to be the same for terms to be like; they only need to share the same variables with the same powers. Recognizing this helps in streamlining the process to combine them later on.

Combining Like Terms

Combining like terms is the process of adding or subtracting them. This step simplifies expressions and makes them easier to work with. To combine like terms, simply perform the arithmetic operation indicated on their coefficients while keeping the common variable part unchanged.

In our example, the like terms \(3x\) and \(10x\) are added together by adding their coefficients:

• Combine \(3x + 10x\) to get \(13x\).
• Do the same for the constant terms: combine \(-1 - 6\) to get \(-7\).

Each set of like terms is simplified to a single term, making it easier to deal with the expression as a whole. Note how important it is to keep the sign of each term in mind, especially when dealing with negative numbers.

Simplifying Expressions

Simplifying expressions means writing them in a simpler form without changing their value. This involves combining any like terms to produce an expression that is easier to understand or to solve in further calculations.

For the expression given \((3x - 1) + (10x - 6)\), after identifying and combining the like terms, we write it as a simplified expression:

• For \(x\)-terms: \(13x\).
• For constant terms: \(-7\).
• The simplified expression becomes: \(13x - 7\).

Simplifying expressions not only streamlines your work but also helps keep mathematical calculations more organized and manageable. Remember, the goal of simplifying is to create an expression that, although shorter or more concise, retains all the components' original meanings.