Problem 18 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 7 n-3 r, \text { if } n=-6 \text { and } r=2 $$

Step-by-Step Solution

Answer

The value is -48.

Step 1: Identify the Expression and Variables

The expression given is \(7n - 3r\). The values for the variables are \(n = -6\) and \(r = 2\).

Step 2: Substitute the Variables into the Expression

Replace \(n\) with \(-6\) and \(r\) with \(2\) in the expression: \(7(-6) - 3(2)\).

Step 3: Calculate the Term Involving \(n\)

Calculate \(7 \times (-6)\), which gives \(-42\).

Step 4: Calculate the Term Involving \(r\)

Calculate \(3 \times 2\), which gives \(6\).

Step 5: Combine the Results

Subtract the term involving \(r\) from the term involving \(n\): \(-42 - 6\).

Step 6: Final Calculation

Perform the final subtraction: \(-42 - 6 = -48\).

Key Concepts

Variable Substitution

When you face an algebraic expression, one of the first things you might need to do is substitute the variables with actual numbers. This process is called variable substitution. Start by identifying the variables in the expression. Just like in the given exercise, where the expression is \(7n - 3r\), the variables are \(n\) and \(r\). Once you know what numbers you will use for each variable—here, \(n = -6\) and \(r = 2\)—replace the variables in the expression with these numbers. So, replace \(n\) with \(-6\) and \(r\) with \(2\), resulting in the new expression:

• \(7(-6) - 3(2)\)

Performing these replacements is a fundamental step to transition from an algebraic expression to a mathematical operation. It’s like opening a door to begin evaluating the expression properly.

Expression Evaluation

Once you've substituted the variables with their respective values, the next step is to evaluate the expression. This step involves carrying out the arithmetic operations in the modified expression. In our example:

• We first handle the multiplication: \(7 \times (-6)\) equals \(-42\).
• Similarly, for the second part, \(3 \times 2\) equals \(6\).

After completing these calculations, you will have two results that you need to sum up or subtract according to the expression. Here, you will subtract the second result from the first, yielding:

• \(-42 - 6\).

This process is essential to simplify complex expressions into easy-to-handle calculations, enabling you to find the value of the original algebraic expression. Approaching it step by step, focusing on one operation at a time, can significantly ease the task.

Simplifying Expressions

The final step is to simplify the expression down to a single value. Simplifying means performing the final operations that combine all parts of the expression into one number.Once you perform the operations indicated by the expression, bring everything together:

• For \(-42 - 6\), simply subtract \(6\) from \(-42\) to get \(-48\).

Simplification helps to convert the expression into its simplest form, which is often a single number or a very basic expression.In this exercise, you aimed to simplify the expression \(7n - 3r\) by substituting, evaluating, and combining, thus finding its value clearly and straightforwardly. Always remember: simplifying expressions streamlines problem-solving, making it approachable and effortless.