Analyse the 2 student's reasoning and steps compared to the correct teacher answer, catch any mistakes. You have to pick a clear winner, and why. I'll provide you 2 sets of student answers at a time. You'll have to pick the overall best student so far, and the best one from the set.

Problem:
```
| x | y |
|---|---|
| k | 13 |
| k + 7 | -15 |

The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is (k - 5, b), where *k* and *b* are constants. What is the value of *b*?
```

Correct teacher answer and reasoning:
```
**Rationale**

The correct answer is 33. It's given in the table that the coordinates of two points on a line in the xy-plane are (k, 13) and (k + 7, -15). The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, _m_, between any two points, (x<sub>1</sub>, y<sub>1</sub>) and (x<sub>2</sub>, y<sub>2</sub>), on the line can be calculated using the slope formula, m = (y<sub>2</sub> - y<sub>1</sub>) / (x<sub>2</sub> - x<sub>1</sub>). 

It follows that the slope of the line with the given points from the table, (k, 13) and (k + 7, -15), is m = (-15 - 13) / (k + 7 - k), which is equivalent to m = -28 / 7, or m = -4. 

It's given that the y-intercept of the line is (k - 5, b). Substituting -4 for _m_ and the coordinates of the points (k - 5, b) and (k, 13) into the slope formula yields -4 = (13 - b) / (k - (k - 5)), which is equivalent to -4 = (13 - b) / 5, or -4 = (13 - b) / 5. 

Multiplying both sides of this equation by 5 yields -20 = 13 - b. Subtracting 13 from both sides of this equation yields -33 = -b. Dividing both sides of this equation by -1 yields b = 33. Therefore, the value of _b_ is 33.
```

Student #1:
```
To find the value of *b*, we need to find the equation of the line first. We can use the two given points to find the slope of the line:

Slope = (y2 - y1) / (x2 - x1)
= (-15 - 13) / ((k + 7) - k)
= (-15 - 13) / 7
= -28 / 7
= -4

Now we can use the slope and one of the points to find the equation of the line. We'll use the point (k, 13):

y - 13 = -4(x - k)

Simplify the equation:

y - 13 = -4x + 4k
y = -4x + 4k + 13

Now we know the y-intercept is (k - 5, b). To find *b*, we can plug in x = k - 5 into the equation of the line:

b = -4(k - 5) + 4k + 13
= -4k + 20 + 4k + 13
= 20 + 13
= 33

So, the value of *b* is 33.
```

Student #2:
```
Here's how to solve this problem:

1. **Find the slope:**  The slope of a line is given by the change in y divided by the change in x. Using the two points in the table:

   Slope = (y2 - y1) / (x2 - x1) = (-15 - 13) / (k + 7 - k) = -28 / 7 = -4

2. **Use the slope-intercept form:** The equation of a line can be written in slope-intercept form as y = mx + b, where m is the slope and b is the y-intercept. We know the slope (m) is -4.

3. **Plug in a point to find the y-intercept:** We can use either of the points from the table. Let's use (k, 13):

   13 = -4(k) + b

4. **Use the y-intercept coordinates:** We're given that the y-intercept is (k - 5, b). Remember that the x-coordinate of the y-intercept is always 0. Therefore, k - 5 = 0, which means k = 5.

5. **Solve for b:** Substitute k = 5 into the equation we found in step 3:

   13 = -4(5) + b
   13 = -20 + b
   b = 13 + 20
   b = 33

Therefore, the value of *b* is 33.
```