The equation of a line passing through the points A (-1,1)and B (2, - 4) is

Options

A. 3x+5y+2=0
B. 5x +3y+2 =0
C. 2x+3y+5=0
D. none to these

Answer

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

### Step 1: Identify the coordinates of the points
The coordinates of point A are (-1, 1) which means \( x_1 = -1 \) and \( y_1 = 1 \). The coordinates of point B are (2, -4) which means \( x_2 = 2 \) and \( y_2 = -4 \).

### Step 2: Calculate the slope (m) of the line
The formula for the slope \( m \) of a line passing through 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 equation of a line
The point-slope form of the equation of a line is given by:

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

Substituting \( m = -\frac{5}{3} \), \( x_1 = -1 \), and \( y_1 = 1 \):

\[ y - 1 = -\frac{5}{3}(x - (-1)) \]

This simplifies to:

\[ y - 1 = -\frac{5}{3}(x + 1) \]

### Step 4: Distribute and simplify the equation
Distributing the slope on the right side:

\[ y - 1 = -\frac{5}{3}x - \frac{5}{3} \]

Now, add 1 to both sides:

\[ y = -\frac{5}{3}x - \frac{5}{3} + 1 \]

To combine the constants, convert 1 to a fraction:

\[ 1 = \frac{3}{3} \]

So:

\[ y = -\frac{5}{3}x - \frac{5}{3} + \frac{3}{3} = -\frac{5}{3}x - \frac{2}{3} \]

### Step 5: Rearranging to standard form
To convert this to standard form \( Ax + By + C = 0 \):

Multiply through by 3 to eliminate the fractions:

\[ 3y = -5x - 2 \]

Rearranging gives:

\[ 5x + 3y + 2 = 0 \]

### Final Answer
The equation of the line passing through points A and B is:

\[ 5x + 3y + 2 = 0 \]

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