Equation of line joining (1, -1) and (3, 5) is

Question

Accepted Answer

To determine the equation of the line joining the points (1, -1) and (3, 5), we need to find the slope of the line and use the point-slope form of the equation of a line.

1. **Calculate the slope (m):**

The formula for the slope $ m $ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
   $$
   m = \frac{y_2 - y_1}{x_2 - x_1}
   $$

Substituting the given points (1, -1) and (3, 5):
   $$
   m = \frac{5 - (-1)}{3 - 1} = \frac{5 + 1}{2} = \frac{6}{2} = 3
   $$

2. **Use the point-slope form:**

The point-slope form of a line is:
   $$
   y - y_1 = m(x - x_1)
   $$

Using the point (1, -1) and the slope $ m = 3 $:
   $$
   y + 1 = 3(x - 1)
   $$

Simplifying:
   $$
   y + 1 = 3x - 3
   $$
   $$
   y = 3x - 3 - 1
   $$
   $$
   y = 3x - 4
   $$

3. **Convert to standard form:**

The standard form of a line is $ Ax + By + C = 0 $. Rearranging $ y = 3x - 4 $ gives:
   $$
   3x - y - 4 = 0
   $$

This can be rewritten as:
   $$
   -3x + y + 4 = 0
   $$

This matches Option 2: $-3x + y + 4 = 0$.

**Explanation of why other options are incorrect:**

- **Option 1: $3x - y + 4 = 0$**
  - This equation implies a different slope and intercept, which does not match the calculated slope of 3 and the derived equation.

- **Option 3: $x - 3y + 4 = 0$**
  - This equation rearranges to $x = 3y - 4$, which implies a slope of $\frac{1}{3}$, not matching the calculated slope of 3.

- **Option 4: $x + 3y + 4 = 0$**
  - This equation rearranges to $x = -3y - 4$, which implies a slope of $-\frac{1}{3}$, again not matching the calculated slope of 3.

Therefore, the correct equation of the line joining the points (1, -1) and (3, 5) is indeed Option 2: $-3x + y + 4 = 0$.