Problem 78 - Elementary Algebra

Question

Find the value of each of the following expressions. \(P=R-C . \quad\) Find \(P\) if \(R=2000\) and \(C=2500\).

Step-by-Step Solution

Answer

Answer: The value of P is -500.

Step 1: Write down the given expression

The given expression is: \(P = R - C\)

Step 2: Substitute the given values of R and C

We must now substitute the given values for \(R\) and \(C\) into the expression. Given that \(R=2000\) and \(C=2500\), we can substitute these values into the expression: \(P = 2000 - 2500\)

Step 3: Calculate P

Now, we just need to evaluate the expression after substituting the values to find \(P\). \(P = 2000 - 2500 = -500\) So, the value of \(P\) is \(-500\).

Key Concepts

Expressions

In algebra, an expression is a combination of numbers, variables, and arithmetic operations, like addition, subtraction, multiplication, or division. Expressions can include constants (fixed numbers) and variables (symbols that represent numbers). For example, in the expression \( P = R - C \), \( P \), \( R \), and \( C \) are variables. This particular expression represents a relationship where \( P \) equals \( R \) minus \( C \).

Algebraic expressions serve as fundamental building blocks in algebra and are crucial for forming equations and inequalities. Becoming comfortable with expressions is essential as it allows you to solve for unknowns and understand mathematical relationships. To work with them, you often need to substitute specific values in place of the variables, turning expressions into numerical calculations.

Substitution

Substitution in algebra involves replacing variables in an expression with known values. For example, if you know \( R = 2000 \) and \( C = 2500 \) in the expression \( P = R - C \), you substitute these values directly into the expression: \( P = 2000 - 2500 \). This changes the expression from a generalized formula to a specific numerical computation.

This process helps simplify expressions, making them easier to evaluate. Substitution is not just about plugging numbers in; it involves understanding how inserting different values can change the shape or outcome of an expression. It allows you to apply abstract mathematical rules to real-world numbers, revealing meaningful results or solutions.

Negative Numbers

Negative numbers are numbers less than zero, indicated by a minus sign \((-\)). They can often be a point of confusion but are essential in various calculations, particularly in expressions involving subtraction or determining a loss. In our example, when \( R = 2000 \) and \( C = 2500 \), substituting these values gives us \( P = 2000 - 2500 \), resulting in a negative number \( P = -500 \).

Adjusting to handling negative numbers is important. They follow the same arithmetic rules as positive numbers, but when subtracting a larger number from a smaller one, the result is negative. This can represent situations such as owing money or other similar contexts where quantities decrease below zero. Arithmetic involving negative numbers is crucial for understanding real-world scenarios and interpreting mathematic phenomena accurately.