Find an equation of the line that satisfies the given conditions.
Through 
(−5, 3);
  perpendicular to the line 
y = − 
1
3
 x + 5

To find the equation of a line that passes through the point \((-5, 3)\) and is perpendicular to the line given by the equation \(y = -\frac{1}{3}x + 5\), we can follow these steps:

### Step 1: Identify the slope of the given line.
The equation of the given line is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope. For the given line \(y = -\frac{1}{3}x + 5\), the slope \(m\) is \(-\frac{1}{3}\).

### Step 2: Find the slope of the perpendicular line.
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. The negative reciprocal of \(-\frac{1}{3}\) is \(3\). Therefore, the slope of the line we want to find is \(m' = 3\).

### Step 3: Use the point-slope form to find the equation.
The point-slope form of a line is given by:
\[
y - y_1 = m'(x - x_1)
\]
where \((x_1, y_1)\) is the point the line passes through, and \(m'\) is the slope of the line.

Substituting the point \((-5, 3)\) and the slope \(m' = 3\) into the equation:
\[
y - 3 = 3(x - (-5))
\]
\[
y - 3 = 3(x + 5)
\]
\[
y - 3 = 3x + 15
\]

### Step 4: Simplify the equation.
Add 3 to both sides to put the equation in slope-intercept form:
\[
y = 3x + 18
\]

So, the equation of the line that passes through \((-5, 3)\) and is perpendicular to the line \(y = -\frac{1}{3}x + 5\) is:
\[
\boxed{y = 3x + 18}
\]

Would you like further details on this solution or have any other questions?

Here are some related questions for further practice:
1. Find the equation of a line parallel to \(y = 2x - 4\) and passing through the point \((1, 2)\).
2. Determine the slope of a line that is perpendicular to \(y = 4x + 7\).
3. What is the equation of a line that passes through \((2, -1)\) and is parallel to \(y = \frac{1}{2}x - 3\)?
4. Calculate the point of intersection between the lines \(y = 3x + 18\) and \(y = -\frac{1}{3}x + 5\).
5. Find the slope-intercept form of a line passing through \((0, 0)\) and perpendicular to \(y = 7x + 2\).

**Tip:** When finding the slope of a perpendicular line, always take the negative reciprocal of the original line's slope.

Learn how to find the equation of a line that passes through the point (-5, 3) and is perpendicular to the line y = -1/3x + 5. Step-by-step solution using the point-slope form of a line provided.

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