Problem 15 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 4 x+7 x $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(4x + 7x\). Answer: \(11x\)

Step 1: Identifying like terms

Both terms in the expression, \(4x\) and \(7x\), have the variable 'x' raised to the same power (power of \(1\)). So, they are considered as like terms.

Step 2: Combining like terms

When combining like terms, we add the coefficients while keeping the variable and its power the same. So, \((4 + 7) x = 11x\).

Step 3: Writing the simplified expression

The simplified algebraic expression is $$11x$$.

Key Concepts

Understanding Like Terms

In algebra, 'like terms' are terms that have the same variable raised to the same power. Let's break it down.
Consider the terms in our exercise:

• For example, in the expression \[4x + 7x\], both terms have the variable 'x' raised to the power of 1.
• They are like terms because they share the same variable and exponent.

Why is this important? Identifying like terms makes it easier to simplify expressions.
If terms don't share a variable or the same power, they are unlike and cannot be combined in the same way.
For instance, \[5x^2\] and \[3x\] are not like terms, as they have different powers.

Simplifying Expressions

Simplifying expressions is an algebraic process of reducing complexity while maintaining the expression's value.
In our example:

• We start with \[4x + 7x\].
• This involves combining like terms to streamline the expression.

This process effectively reduces the number of terms, making further operations more straightforward.
For a student, learning to simplify expressions can transform a bewildering algebraic problem into a manageable task.
The goal is to achieve an expression where no further simplification is possible, resulting in just one term if available.
Thus, for our task, we end with \[11x\].

Combining Coefficients

Combining coefficients concerns the numbers in front of variables, known as coefficients.
These numbers multiply the variable in an algebraic term.
When like terms are combined, their coefficients can be added or subtracted while keeping the variable intact.
In our example:

• The coefficients are \[4\] and \[7\].
• We add these coefficients to simplify: \[4 + 7 = 11\].

This results in \[11x\], where \[11\] is the new coefficient of the variable.
Combining coefficients correctly results in a clearer and simpler expression.
Recognizing and using this technique is crucial, as it allows for easier manipulation and understanding of algebraic expressions.