The line passing through (0, 2) and (-3, -1) is parallel to the line passing through (-1, 5) and `(4,a)`. Find a. 

Answer

To find the value of \( a \) such that the line passing through the points \( (0, 2) \) and \( (-3, -1) \) is parallel to the line passing through the points \( (-1, 5) \) and \( (4, a) \), we can follow these steps:

### Step 1: Calculate the slope of the first line
The slope \( m_1 \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For the points \( (0, 2) \) and \( (-3, -1) \):
- \( x_1 = 0, y_1 = 2 \)
- \( x_2 = -3, y_2 = -1 \)

Substituting these values into the slope formula:
\[ m_1 = \frac{-1 - 2}{-3 - 0} = \frac{-3}{-3} = 1 \]

### Step 2: Calculate the slope of the second line
Now, we need to calculate the slope \( m_2 \) of the line passing through the points \( (-1, 5) \) and \( (4, a) \):
- \( x_1 = -1, y_1 = 5 \)
- \( x_2 = 4, y_2 = a \)

Using the slope formula:
\[ m_2 = \frac{a - 5}{4 - (-1)} = \frac{a - 5}{4 + 1} = \frac{a - 5}{5} \]

### Step 3: Set the slopes equal
Since the two lines are parallel, their slopes must be equal:
\[ m_1 = m_2 \]
Substituting the values we found:
\[ 1 = \frac{a - 5}{5} \]

### Step 4: Solve for \( a \)
To solve for \( a \), we can cross-multiply:
\[ 1 \cdot 5 = a - 5 \]
This simplifies to:
\[ 5 = a - 5 \]
Adding 5 to both sides gives:
\[ a = 10 \]

### Conclusion
The value of \( a \) is \( 10 \).

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