Problem 27 - Fundamentals of Mathematics

Question

In the expression \(-7 c\), how many \(c\) 's are indicated?

Step-by-Step Solution

Answer

The expression \(-7c\) indicates 7 \(c\)'s.

Step 1: Understand the Expression

We start with the expression \(-7c\). This expression consists of two parts: the coefficient \(-7\) and the variable \(c\).

Step 2: Identify the Coefficient

In the expression \(-7c\), the coefficient is \(-7\). The coefficient tells us how many times the variable \(c\) is being multiplied.

Step 3: Interpret the Coefficient

The coefficient \(-7\) indicates that there are 7 of the variable \(c\), but they are all negative. The negative sign represents the direction, but it does not change the count of \(c\)'s except to indicate their negative nature.

Step 4: Count the Number of c's

Since the coefficient is \(-7\), this means there are 7 units of \(c\). We acknowledge the negative sign, but it does not affect the number of \(c\)'s counted in terms of quantity.

Key Concepts

Coefficients and Variables

Understanding algebraic expressions is much easier when you identify its core components: coefficients and variables. An expression like office -7c is constructed from two main parts:

• Coefficient: This is a numerical factor placed before a variable that scales or affects the size of the variable. In our example, office -7 is the coefficient, showing how many times the variable is taken.
• Variable: Represents an unknown or a placeholder for values that can change. Here, office c stands for the variable.

Recognizing these two parts helps in simplifying and understanding expressions better.
Just keep in mind: coefficients tell you how many times to add or subtract the variables involved.

Multiplication of Variables

When you see a variable attached to a coefficient, like in office -7c the operation implied is multiplication. This means that the variable office c is multiplied by the number office -7. Think of multiplication here as simply scaling or repeating.

• Your expression indicates that you have office c repeated or accumulated office 7 times.
• The coefficient's sign tells the nature of this repetition, whether positive or negative.

So, the multiplication doesn't just combine numbers; it amplifies the variable by the factor given by the coefficient.
Understanding this scaling ability is crucial to mastering algebra.

Interpreting Negative Coefficients

When dealing with negative coefficients in expressions like office -7c, there's an extra layer of meaning. First, the number part of the coefficient still tells you the count, which is 7 in office -7c.

• The negative sign indicates not just subtraction every time office c is added, but that each unit of office c is summed in a negative direction.
• It does not change the number of office c units, just their sign or direction.
• Think of it like owing 7 copies of office c instead of having them positively.

This understated representation allows algebra to express gains and losses through simple expressions.
Mastering this understanding of negative coefficients helps you judge the results of arithmetic operations within equations more intuitively.