Find the equation of a line passing through the points (-1,1) and (2,-4)

Options

A. `5x-3y-2=0`
B. `5x-3y+2=0`
C. `5x+3y+2=0`
D. `5x+6y+2=0`

Explanation

To find the equation of the line passing through the points (-1, 1) and (2, -4), we can use the point-slope form of the equation of a line. The point-slope form is given by:

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

where \( m \) is the slope of the line, and \((x_1, y_1)\) is a point on the line.

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

The slope \( m \) can be calculated using the formula:

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

Substituting the given points \((-1, 1)\) and \((2, -4)\):

- \(x_1 = -1\), \(y_1 = 1\)
- \(x_2 = 2\), \(y_2 = -4\)

Calculating the slope:

\[ m = \frac{-4 - 1}{2 - (-1)} = \frac{-5}{2 + 1} = \frac{-5}{3} \]

### Step 2: Use the point-slope form

Now that we have the slope \( m = -\frac{5}{3} \), we can use the point-slope form with one of the points. We will use the point \((-1, 1)\):

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

This simplifies to:

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

### Step 3: Distribute and simplify

Distributing \(-\frac{5}{3}\):

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

Now, add 1 to both sides:

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

Convert 1 to a fraction with a denominator of 3:

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

Combine the constant terms:

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

### Step 4: Rearranging to standard form

To express this in standard form \(Ax + By + C = 0\):

\[ \frac{5}{3}x + y + \frac{2}{3} = 0 \]

Multiplying through by 3 to eliminate the fractions:

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

### Final Answer

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

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

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