To solve the problem, we need to identify the slopes of the given linear functions and arrange them in increasing order.

The six linear functions are:

1. $a(y = -x - 7)$
2. $b(y = 3x + 1)$
3. $c(y = -10x)$
4. $d(y = x)$
5. $e\left(y = \frac{x}{10} + 7\right)$
6. $f(y = -100)$

Step 1: Identify the slopes of each function.

• For the function $y = mx + c$, the slope is the coefficient $m$ of $x$.

1. $y = -x - 7$: The slope is $-1$.
2. $y = 3x + 1$: The slope is $3$.
3. $y = -10x$: The slope is $-10$.
4. $y = x$: The slope is $1$.
5. $y = \frac{x}{10} + 7$: The slope is $\frac{1}{10}$.
6. $y = -100$: The slope is $0$ (since there's no $x$ term).

Step 2: Arrange the slopes in increasing order.

The slopes from least to greatest are:

• $-10$ (from $c(y = -10x)$)
• $-1$ (from $a(y = -x - 7)$)
• $0$ (from $f(y = -100)$)
• $\frac{1}{10}$ (from $e\left(y = \frac{x}{10} + 7\right)$)
• $1$ (from $d(y = x)$)
• $3$ (from $b(y = 3x + 1)$)

Step 3: Match this order to the given options.

The correct order according to the slope is:

• $c, a, f, e, d, b$

So the correct answer is Option C: (c), (a), (f), (e), (d), (b).

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What is the general form of a linear equation and how do you determine its slope?
2. How does the slope affect the direction and steepness of a line?
3. What happens to the slope when a line is parallel to the x-axis?
4. How can you find the slope between two points on a line?
5. How does the slope-intercept form relate to real-world scenarios?

Tip: Remember that the slope of a line can tell you whether the line is increasing, decreasing, or horizontal.