Problem 6 - Fundamentals of Mathematics

Question

Find the value of each expression. $$9 m-2 n, \text { if } m=-2 \text { and } n=5$$

Step-by-Step Solution

Answer

The value of the expression is -28.

Step 1: Understand the Expression

You need to evaluate the expression \(9m - 2n\) by substituting the given values of \(m\) and \(n\).

Step 2: Substitute the Values

Replace the variable \(m\) with \(-2\) and the variable \(n\) with \(5\) in the expression. This gives: \(9(-2) - 2(5)\).

Step 3: Calculate the Multiplication

Perform the multiplication operations: Calculate \(9 \times (-2) = -18\) and \(-2 \times 5 = -10\).

Step 4: Perform the Addition

Add the results from Step 3: \(-18 + (-10) = -28\).

Step 5: Conclusion

The final value of the expression \(9m - 2n\) when \(m = -2\) and \(n = 5\) is \(-28\).

Key Concepts

Evaluating Expressions

Evaluating expressions involves substituting specific values for variables and then performing the necessary calculations to find the resultant value of the expression. Consider the expression \(9m - 2n\).

• Substitution: To evaluate this algebraic expression, you replace variables with the given numbers. Here, substitute \(m = -2\) and \(n = 5\).
• Calculation: After substitution, follow arithmetic operations to calculate the final value. In our case, substituting the values results in \(9(-2) - 2(5)\).

Evaluating expressions helps in understanding how variables affect the outcomes of an expression. It's very similar to solving a puzzle with numbers replacing pieces.

Order of Operations

The order of operations is fundamental in mathematics to avoid confusion and ensure every expression is evaluated correctly. Here is a simple acronym to remember it: **PEMDAS**

• **P**arentheses first
• **E**xponents (i.e., powers and roots, etc.)
• **MD** Multiplication and Division (from left to right)
• **AS** Addition and Subtraction (from left to right)

Let's apply this to our expression: \(9(-2) - 2(5)\). - First, handle multiplication: \(9 \times (-2)\) equals \(-18\) and \(-2 \times 5\) equals \(-10\).- Then, perform the subtraction since there are no parentheses or exponents here: subtracting \(10\) from \(-18\) results in \(-28\).Following this sequence of operations ensures that every expression evaluates accurately.

Algebraic Expressions

An algebraic expression is a combination of constants, variables, and operation symbols (like +, -, ×, ÷). Unlike equations, they do not have an equality sign. Here are some key components:

• **Variables**: Symbols representing unspecified numbers; here \(m\) and \(n\) are variables.
• **Constants**: Specific numbers that are always the same; in \(9m - 2n\), \(9\) and \(-2\) are constants.
• **Operations**: Mathematical processes like addition, subtraction, etc.

The purpose of forming algebraic expressions is to model and solve real-world problems algebraically. For example, \(9m - 2n\) can represent numerous practical situations, and by substituting values, you can explore these scenarios systematically.