Problem 55 - Prealgebra

Question

Simplify each expression. $$4 x+7 x$$

Step-by-Step Solution

Answer

The expression simplifies to \(11x\).

Step 1: Identify Like Terms

In the expression \(4x + 7x\), identify the like terms that can be combined. Both \(4x\) and \(7x\) are like terms because they both have the variable \(x\).

Step 2: Combine Like Terms

Add the coefficients of the like terms \(4x\) and \(7x\). The coefficients are 4 and 7. So, add them together: \(4 + 7 = 11\).

Step 3: Rewrite the Simplified Expression

Once the coefficients are combined, rewrite the expression using the sum and the common variable. This gives you \(11x\).

Key Concepts

Combining Like Terms

Combining like terms is an essential skill in algebra. It involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. For example, in the expression \(4x + 7x\), both terms contain the variable \(x\). They are considered like terms because the variable and its power are identical.

Why do we combine like terms? Because it makes expressions simpler and easier to understand. Imagine having a basket of apples and oranges. If you want to know how many apples you have, you would only count the apples, right? Similarly, combining like terms helps us to "count" similar items in an algebraic expression.

• Identify terms with the same variable.
• Add or subtract the coefficients of these terms.
• Keep the variable part unchanged.

Thus, by combining \(4x\) and \(7x\), you simply add their coefficients, giving \(11x\).

Simplifying Expressions

Simplifying expressions is a fundamental part of algebra that involves reducing an expression to its simplest form. This process often includes combining like terms, as we've seen. Simplification makes expressions more manageable and easier to work with when solving equations or performing other calculations.

Consider our initial goal: simplify \(4x + 7x\). By combining the like terms, we reduced the expression to \(11x\). When expressions are simplified, it offers a clearer view of the relationship between variables and coefficients. Here’s how to achieve that:

• First, group like terms together, just like sorting similar items in a drawer.
• Then, perform arithmetic actions on the coefficients only, leaving the variables untouched.

This approach results in a much cleaner, more comprehensible expression, setting a solid foundation for further algebraic operations.

Coefficients in Algebra

Coefficients are the numerical part of the terms that accompany the variable in algebraic expressions. They tell us "how many" of the variable we have. In our example, \(4x\) and \(7x\) have coefficients 4 and 7 respectively.

Understanding coefficients is crucial because they provide the quantity factor in expressions. They are the multiplying factor for the variable part of the terms. When you're asked to combine like terms, your job mainly involves the coefficients:

• Add the coefficients if you're adding similar terms.
• Subtract them if the terms involve subtraction.

In conclusion, by understanding coefficients, you can more effectively simplify expressions, as you saw with \(4 + 7 = 11\), leading to the term \(11x\). Recognizing coefficients streamlines the process of manipulating algebraic expressions.