Find the equation of the line that contains the point 
P(−3, 3)
 and is perpendicular to the graph of 
3x + 6y = −5.


To find the equation of the line that passes through the point \( P(-3, 3) \) and is perpendicular to the graph of the line given by the equation \( 3x + 6y = -5 \), we will follow these steps:

### Step 1: Find the slope of the given line.

The given line is:

\[
3x + 6y = -5
\]

First, we convert this equation to the slope-intercept form \( y = mx + b \), where \( m \) is the slope.

\[
6y = -3x - 5
\]

\[
y = -\frac{1}{2}x - \frac{5}{6}
\]

From this equation, we see that the slope of the given line is \( m = -\frac{1}{2} \).

### Step 2: Determine the slope of the line perpendicular to the given line.

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Thus, if the slope of the original line is \( m = -\frac{1}{2} \), then the slope of the perpendicular line \( m_{\perp} \) is:

\[
m_{\perp} = -\left(-\frac{1}{2}\right)^{-1} = 2
\]

### Step 3: Write the equation of the line with the perpendicular slope passing through point \( P(-3, 3) \).

Now that we have the slope \( m_{\perp} = 2 \), we can use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

where \( (x_1, y_1) \) is the point \( P(-3, 3) \) and \( m = 2 \).

Substitute these values into the point-slope formula:

\[
y - 3 = 2(x + 3)
\]

### Step 4: Simplify the equation.

Expand and simplify to get the equation in slope-intercept form:

\[
y - 3 = 2x + 6
\]

\[
y = 2x + 9
\]

### Final Answer:

The equation of the line that passes through the point \( P(-3, 3) \) and is perpendicular to the graph of \( 3x + 6y = -5 \) is:

\[
\boxed{y = 2x + 9}
\]

Would you like more details or have any other questions?

Here are 5 related questions for you to consider:

1. How do you find the equation of a line parallel to a given line?
2. How can you determine if two lines are perpendicular just by looking at their equations?
3. What is the point-slope form of a linear equation, and when is it useful?
4. How do you find the distance between a point and a line?
5. How do you convert a line equation from standard form to slope-intercept form?

**Tip:** To find the slope of any line, first try to rearrange the equation into the slope-intercept form \( y = mx + b \), where \( m \) is the slope!

Find the equation of the line that contains the point P(−3, 3) and is perpendicular to the graph of 3x + 6y = −5.

Learn how to find the equation of the line that passes through the point P(-3, 3) and is perpendicular to the line 3x + 6y = -5. This step-by-step solution explains key algebraic concepts like slope, perpendicular slopes, and how to use point-slope form. Suitable for high school students in Grades 9-12.

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