Find the equation of the line passing through the points (–1, 1) and (2, –4).
(a)5x+3y+2=0
(b)3x+6y=9
(c)4x-6y-8=0
(d)None of these

Answer

To find the equation of the line passing through the points (-1, 1) and (2, -4), we can follow these steps:

### Step 1: Identify the points
We have two points:
- Point 1: \( (x_1, y_1) = (-1, 1) \)
- Point 2: \( (x_2, y_2) = (2, -4) \)

### Step 2: Calculate the slope (m)
The formula for the slope \( m \) between two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values:
\[ m = \frac{-4 - 1}{2 - (-1)} = \frac{-5}{2 + 1} = \frac{-5}{3} \]

### Step 3: Use the point-slope form of the line equation
The point-slope form of the line is:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m \), \( x_1 \), and \( y_1 \):
\[ y - 1 = -\frac{5}{3}(x - (-1)) \]
This simplifies to:
\[ y - 1 = -\frac{5}{3}(x + 1) \]

### Step 4: Distribute and rearrange the equation
Distributing \( -\frac{5}{3} \):
\[ y - 1 = -\frac{5}{3}x - \frac{5}{3} \]
Adding 1 to both sides:
\[ y = -\frac{5}{3}x - \frac{5}{3} + 1 \]
Converting 1 to a fraction:
\[ y = -\frac{5}{3}x - \frac{5}{3} + \frac{3}{3} = -\frac{5}{3}x - \frac{2}{3} \]

### Step 5: Convert to standard form
To convert to standard form \( Ax + By + C = 0 \), we can multiply through by 3 to eliminate the fractions:
\[ 3y = -5x - 2 \]
Rearranging gives:
\[ 5x + 3y + 2 = 0 \]

### Conclusion
The equation of the line passing through the points (-1, 1) and (2, -4) is:
\[ 5x + 3y + 2 = 0 \]
Thus, the correct answer is option (a).

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