Problem 8 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$6 h-2 h$$

Step-by-Step Solution

Answer

The expression simplifies to \(4h\).

Step 1: Identify Like Terms

Begin by examining the expression \(6h - 2h\). The terms are considered 'like terms' because they both contain the variable \(h\) raised to the same power (which is 1). This means we can combine them.

Step 2: Apply the Properties of Subtraction

When combining like terms, apply the properties of subtraction. In this expression, you will subtract the coefficients of the like terms. The coefficients in our terms are 6 and 2.

Step 3: Subtract the Coefficients

Perform the subtraction on the coefficients: \(6 - 2 = 4\). This subtraction results in a new coefficient for the variable \(h\).

Step 4: Write the Simplified Expression

Combine the result from the coefficient subtraction with the variable. The simplified expression is \(4h\).

Key Concepts

Simplifying Algebraic Expressions

Algebraic expressions can become cluttered with numerous terms, making them hard to interpret. To simplify them, you combine terms that are "like." Like terms have the exact same variable part, which means the variable itself and its exponent must match. This enables simplification because it reduces the number of terms, simplifying calculations and making the expression easier to work with. When dealing with expressions, there are some helpful steps to follow:

• Identify terms that have the same variable parts.
• Combine them by performing operations on the coefficients.

By following these steps, you simplify expressions to their most basic form.

Coefficients in Algebra

In algebraic expressions, each term has a coefficient, a numerical or constant factor that multiplies the variable part. Consider the expression before simplification, such as in the term \(6h\), here 6 is the coefficient. Coefficients tell you how many times you multiply the variable.When combining like terms:

• Focus on the operations between coefficients.
• Leave the variable part the same – only the coefficients change.

You act directly on coefficients since they are the numerical values involved in addition, subtraction, and other operations. Proper handling of coefficients is crucial for correct simplification results in algebra.

Properties of Subtraction

Subtraction in algebra involves reducing one number by another, which can apply to terms within expressions. Understanding these properties empowers you to systematically break down and solve problems.In the context of combining like terms, the properties are:

• Only coefficients of like terms are subtracted.
• The variable remains intact while coefficients are modified.

For example, in \(6h - 2h\), the subtraction operation is only on the coefficients (6 and 2), resulting in the new expression \(4h\). This illustrates acting on coefficients while respecting the integrity of the variables.