Problem 17 - Elementary Algebra

Question

\(-\frac{3}{5} p+\frac{1}{8}-\frac{1}{5} p+\frac{3}{8}\)

Step-by-Step Solution

Answer

o-\frac{4}{5}p + \(\frac{1}{2}\).

Step 1: Combine Like Terms

Identify and combine the like terms that involve the variable, and the constants separately. In this expression, the like terms involving the variable are \( -\frac{3}{5}p \) and \( -\frac{1}{5}p \). The constants are \( \frac{1}{8} \) and \( \frac{3}{8} \).

Step 2: Simplify Variable Terms

Add \( -\frac{3}{5}p \) and \( -\frac{1}{5}p \) together. This results in \(-\frac{3}{5}p - \frac{1}{5}p = -\frac{4}{5}p\).

Step 3: Simplify Constant Terms

Add the constants \( \frac{1}{8} \) and \( \frac{3}{8} \) together. This yields \( \frac{1}{8} + \frac{3}{8} = \frac{4}{8} = \frac{1}{2} \).

Step 4: Combine Results

Combine the simplified variable term and constant term. The result is \( -\frac{4}{5}p + \frac{1}{2} \).

Key Concepts

Elementary Algebra

Elementary algebra forms the foundation of all advanced mathematics, because it introduces the use of letters (or symbols) to represent numbers. It allows us to work with expressions and equations more generally, solving a wide variety of problems.

In algebra, you deal with expressions (like \(-x + y\)), equations (like \(2x + 3 = 5\)), and sometimes, inequalities (like \(3y > 2x\)). Algebreic terms can be either constants or variables.

Understanding the processes of combining like terms, simplifying expressions, and solving equations is crucial for better grasp of elementary algebra.

In our given exercise, we're combining like terms to simplify an expression with both constants and a variable term.

Variable Terms

A variable term in algebraic expressions involves a variable (often represented by letters such as x, y, or p) multiplied by a number. This number is referred to as the coefficient (e.g., in \(3x\), 3 is the coefficient and x is the variable).

### Identifying Variable Terms

• Look for terms with the same variable (e.g., \(-\frac{3}{5}p\) and \(-\frac{1}{5}p\))
• Combine their coefficients by addition or subtraction.

### Example
For the coefficients -\frac{3}{5} and -\frac{1}{5}, combine them as follows:

\(-\frac{3}{5}p\) + \(-\frac{1}{5}p\) = \(-\frac{4}{5}p\)

Constant Terms

Constant terms are the numbers in an expression that do not involve a variable. They stand alone without any letter attached to them. For example, in the expression \(-\frac{3}{5}p + \frac{1}{8} - \frac{1}{5}p + \frac{3}{8}\), \(\frac{1}{8}\) and \(\frac{3}{8}\) are the constant terms.

### Adding Constant Terms
Combine them like regular numbers since they don't involve a variable.

• Add \(\frac{1}{8}\) and \(\frac{3}{8}\) to get \(\frac{4}{8}\) which simplifies to \(\frac{1}{2}\).

This process is straightforward since it involves simple fraction addition.

Simplification

Simplification in algebra means reducing an expression to its simplest form while retaining the equality. It helps to make computations easier and the expressions more understandable.

### Steps to Simplify

• Identify and combine like terms first.
• Handle variable and constant terms separately.
• Combine the results from previous steps to get the simplest form of the expression.

In our exercise, we:

• Combined variable terms \( -\frac{3}{5}p \) and \( -\frac{1}{5}p \) to get \( -\frac{4}{5}p \).
Your results from previous steps to get the final simplified expression:

\[ -\frac{4}{5}p + \frac{1}{2} \]

Simplification makes handling algebraic expressions easier and helps in solving equations effectively.