To find the equation of a line perpendicular to $3x - y = -3$ and passing through the point $(2, -3)$, we can follow these steps:

Step 1: Rewrite the Given Line in Slope-Intercept Form

The equation $3x - y = -3$ can be rewritten as: $y = 3x + 3$ From this form, we see that the slope of the given line is $3$.

Step 2: Find the Perpendicular Slope

The slope of a line perpendicular to one with slope $m$ is the negative reciprocal, so: $\text{Perpendicular slope} = -\frac{1}{3}$

Step 3: Use the Point-Slope Formula

Now, using the point-slope form $y - y_1 = m(x - x_1)$ with point $(2, -3)$ and slope $-\frac{1}{3}$: $y + 3 = -\frac{1}{3}(x - 2)$

Step 4: Simplify

Expanding this equation: $y + 3 = -\frac{1}{3}x + \frac{2}{3}$ Subtract $3$ from both sides: $y = -\frac{1}{3}x - \frac{7}{3}$

Final Answer

The equation of the line perpendicular to $3x - y = -3$ and passing through $(2, -3)$ is: $y = -\frac{1}{3}x - \frac{7}{3}$

Would you like more details on any of these steps?

Relative Questions

1. How do you find the slope of a line given in standard form?
2. What is the importance of the negative reciprocal when finding a perpendicular line?
3. How do you verify if two lines are perpendicular?
4. Can the equation of a line be represented in multiple forms?
5. How do you find the point of intersection between two lines?

Tip:

To quickly find perpendicular slopes, remember to take the reciprocal of the original slope and then change the sign.