write the equation of a line that is perpendicular to the following and passes through (2,-3) 3x-y=-3

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.

Write the equation of a line that is perpendicular to the following and passes through (2, -3): 3x - y = -3

Learn to write the equation of a line perpendicular to 3x - y = -3 that passes through the point (2, -3). This problem uses the point-slope formula and the concept of perpendicular slopes. Ideal for students in Grades 8-10.

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