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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)**.

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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.

Six linear functions are given below: 
1. y = -x - 7 
2. y = 3x + 1 
3. y = -10x 
4. y = x 
5. y = x/10 + 7 
6. y = -100 
If the functions are put in the order of increasing slope, then the correct order is: 
A) (a), (b), (d), (e), (f), (c) 
B) (c), (f), (a), (d), (e), (b) 
C) (c), (a), (f), (e), (d), (b) 
D) (a), (f), (e), (b), (d), (c)

Learn how to arrange linear functions in order of increasing slope. This problem involves understanding the slopes of six linear equations: y = -x - 7, y = 3x + 1, y = -10x, y = x, y = x/10 + 7, and y = -100. Suitable for students in Grades 8-10.

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