Problem 73 - Prealgebra

Question

Simplify. $$4 x-11+3 x$$

Step-by-Step Solution

Answer

The simplified expression is \(7x - 11\).

Step 1: Identify Like Terms

Observe that the terms in the expression are given as \(4x\), \(-11\) (a constant), and \(3x\). Identify that \(4x\) and \(3x\) are like terms because they both contain the variable \(x\).

Step 2: Combine Like Terms

Add the coefficients of the like terms \(4x\) and \(3x\). This involves adding \(4\) and \(3\) to get the coefficient for \(x\): \(4 + 3 = 7\). The combined term is \(7x\).

Step 3: Write Combined Expression

Replace the original terms with the simplified terms. Since we combined \(4x\) and \(3x\) into \(7x\), the expression becomes \(7x - 11\).

Key Concepts

Understanding Like Terms

In algebra, understanding how to identify like terms is crucial for simplifying expressions. Like terms are terms that have the same variables raised to the same power. This means that both the variable(s) and the exponent(s) need to be identical. In the problem at hand, we have terms like \(4x\) and \(3x\). Both of these terms contain the variable \(x\) raised to the power of 1. Thus, they are considered like terms. On the other hand, terms such as \(-11\), which is a constant, do not have a variable part and cannot be grouped with terms that do.

Combining Like Terms

Once like terms are identified, the next step is to combine them to simplify the expression. Combining like terms involves adding or subtracting their coefficients while keeping their common variable and its exponent the same. When we look at the terms \(4x\) and \(3x\) in our exercise:

• The coefficients are 4 and 3, respectively.
• To combine them, simply add the coefficients: \(4 + 3 = 7\).
• Retain the common variable \(x\) leading to the term \(7x\).

Combining like terms reduces the number of terms in an expression, making it easier to work with in further calculations.

Exploring Coefficients

Coefficients are the numerical part of a term that is multiplied by the variable. In the expression we analyzed, both \(4x\) and \(3x\) have coefficients, where 4 and 3 are the coefficients, and \(x\) is the variable.

• The coefficient tells us how many times the variable is multiplied.
• When combining like terms, you only add or subtract the coefficients, not the variables.
• For example, adding coefficients 4 and 3 gives us a new coefficient of 7 for \(x\), resulting in \(7x\).

Understanding coefficients is key to simplifying expressions correctly as it helps indicate the magnitude associated with variables.