Finding Straight Line Equation

To find the equation of a straight line passing through two points, $(x_1, y_1)$ and $(x_2, y_2)$, we first calculate the slope ($m$) and then use the point-slope form.

Calculate Slope

Given points are $(-2, 5)$ and $(1, 3)$.

Let $(x_1, y_1) = (-2, 5)$ and $(x_2, y_2) = (1, 3)$.

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Substituting the values:

$m = \frac{3 - 5}{1 - (-2)} = \frac{-2}{1 + 2} = \frac{-2}{3}$

Apply Point-Slope Form

The point-slope form of a line is $y - y_1 = m(x - x_1)$. We can use either point. Let's use $(1, 3)$.

$y - 3 = -\frac{2}{3}(x - 1)$

Simplify Equation

Multiply both sides by 3 to eliminate the fraction:

$3(y - 3) = -2(x - 1)$

$3y - 9 = -2x + 2$

Rearrange the terms to match the standard form $Ax + By + C = 0$:

$2x + 3y - 9 - 2 = 0$

$2x + 3y - 11 = 0$

Verify Result

The equation $2x + 3y - 11 = 0$ corresponds to Option 3.

Let's check if the other point $(-2, 5)$ satisfies this equation:

$2(-2) + 3(5) - 11 = -4 + 15 - 11 = 11 - 11 = 0$

The equation holds true for both points.