Analyse the 2 student's reasoning and steps compared to the correct teacher answer, catch any mistakes.

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 \( b \), which represents the y-intercept of the line, we need to determine the equation of the line first. We have two points given:

- Point 1: \((k, 13)\)
- Point 2: \((k + 7, -15)\)

### Step 1: Calculate the slope \( m \)

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the given points:

\[ m = \frac{-15 - 13}{(k + 7) - k} = \frac{-15 - 13}{7} = \frac{-28}{7} = -4 \]

### Step 2: Use the point-slope form to find the equation of the line

The point-slope form of the equation of a line is:

\[ y - y_1 = m(x - x_1) \]

Using point \((k, 13)\):

\[ y - 13 = -4(x - k) \]

### Step 3: Convert to slope-intercept form

The slope-intercept form of a line is:

\[ y = mx + b \]

Expanding and rearranging the equation:

\[ y - 13 = -4x + 4k \]
\[ y = -4x + 4k + 13 \]

### Step 4: Determine the y-intercept

The y-intercept occurs where \( x = 0 \). From the equation:

\[ y = 4k + 13 \]

### Step 5: Use the given y-intercept form

We are given that the y-intercept is at \((k - 5, b)\). This means when \( x = k - 5 \), \( y = b \). Substituting \( x = k - 5 \) into the line equation:

\[ b = -4(k - 5) + 4k + 13 \]

Simplify this equation:

\[ b = -4k + 20 + 4k + 13 \]
\[ b = 33 \]

Thus, the value of \( b \) is:

\[ \boxed{33} \]
```

Student #2:
```
Let the two points be $(x_1, y_1) = (k, 13)$ and $(x_2, y_2) = (k+7, -15)$.
The slope of the line passing through these two points is given by:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-15 - 13}{(k+7) - k} = \frac{-28}{7} = -4$$
The equation of the line can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
We know that the slope is $m = -4$.
Using the point $(k, 13)$, we can write the equation as:
$$13 = -4k + b$$
Using the point $(k+7, -15)$, we can write the equation as:
$$-15 = -4(k+7) + b$$
$$-15 = -4k - 28 + b$$
Now we have a system of two equations with two variables:
1) $13 = -4k + b$
2) $-15 = -4k - 28 + b$
We can solve for $b$ by subtracting equation (2) from equation (1):
$$13 - (-15) = (-4k + b) - (-4k - 28 + b)$$
$$13 + 15 = -4k + b + 4k + 28 - b$$
$$28 = 28$$
This doesn't help us find $b$. Let's try another approach.
We can solve for $b$ in equation (1):
$$b = 13 + 4k$$
Substitute this into equation (2):
$$-15 = -4k - 28 + (13 + 4k)$$
$$-15 = -4k - 28 + 13 + 4k$$
$$-15 = -15$$
This also doesn't help.
We are given that the y-intercept is $(k-5, b)$. This means that when $x = k-5$, $y = b$.
We can use the equation $y = -4x + b$ and plug in the point $(k-5, b)$:
$$b = -4(k-5) + b$$
$$b = -4k + 20 + b$$
$$0 = -4k + 20$$
$$4k = 20$$
$$k = 5$$
Now we can find $b$ using the equation $b = 13 + 4k$:
$$b = 13 + 4(5)$$
$$b = 13 + 20$$
$$b = 33$$

Thus, the value of $b$ is 33.

The y-intercept is $(k-5, b) = (5-5, 33) = (0, 33)$.

The value of $b$ is 33.

Thus, the value of $b$ is 33.

The y-intercept is $(k-5, b) = (5-5, 33) = (0, 33)$.

The value of $b$ is 33.

Thus, the value of $b$ is 33.

The y-intercept is $(k-5, b) = (5-5, 33) = (0, 33)$.

The value of $b$ is 33.

Thus, the value of $b$ is 33.

The y-intercept is $(k-5, b) = (5-5, 33) = (0, 33)$.

The value of $b$ is 33.

Thus, the value of $b$ is 33.

The y-intercept is $(k-5, b) = (5-5, 33) = (0, 33)$.

The value of $b$ is $\boxed{33}$.

Final Answer: The final answer is 33.
```