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Part 1
Find the​ slope-intercept form of the equation of the line that passes through the point Pequals​(negative 9​,negative 8​) and makes angle thetaequals150degrees with the positive​ x-axis.

To find the slope-intercept form of the equation of the line that passes through the point \( P(-9, -8) \) and makes an angle \(\theta = 150^\circ\) with the positive x-axis, follow these steps:

### Step 1: Determine the slope
The slope \( m \) of a line that makes an angle \(\theta\) with the positive x-axis can be found using the tangent function:
\[ m = \tan(\theta) \]

For \(\theta = 150^\circ\):
\[ m = \tan(150^\circ) \]

Using the trigonometric identity:
\[ \tan(150^\circ) = \tan(180^\circ - 30^\circ) = -\tan(30^\circ) \]
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

So,
\[ \tan(150^\circ) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \]

Thus, the slope \( m \) is:
\[ m = -\frac{\sqrt{3}}{3} \]

### Step 2: Use the point-slope form to find the equation of the line
The point-slope form of the equation of a line is:
\[ y - y_1 = m(x - x_1) \]

Where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. For the point \( P(-9, -8) \):
\[ y - (-8) = -\frac{\sqrt{3}}{3}(x - (-9)) \]

Simplify the equation:
\[ y + 8 = -\frac{\sqrt{3}}{3}(x + 9) \]

### Step 3: Convert to slope-intercept form \( y = mx + b \)
Distribute the slope on the right side:
\[ y + 8 = -\frac{\sqrt{3}}{3}x - \frac{\sqrt{3}}{3} \cdot 9 \]

Simplify further:
\[ y + 8 = -\frac{\sqrt{3}}{3}x - 3\sqrt{3} \]

Subtract 8 from both sides to solve for \( y \):
\[ y = -\frac{\sqrt{3}}{3}x - 3\sqrt{3} - 8 \]

Thus, the slope-intercept form of the equation of the line is:
\[ y = -\frac{\sqrt{3}}{3}x - 3\sqrt{3} - 8 \]

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

### Relative Questions:
1. How can we derive the tangent of an angle using trigonometric identities?
2. What are the properties of lines with angles greater than \(90^\circ\) with respect to the x-axis?
3. How does changing the point through which the line passes affect the slope-intercept form?
4. Can we use a different method to find the equation of the line, such as using vectors?
5. What are the implications of having a negative slope for the line's graph?

### Tip:
When working with angles and slopes, remember that the tangent function is periodic with a period of \(180^\circ\), which helps in determining the slope for various angles.

Find the slope-intercept form of the equation of the line that passes through the point P(-9, -8) and makes angle θ = 150° with the positive x-axis.

Learn how to find the slope-intercept form of a line that passes through the point (-9, -8) and makes a 150° angle with the x-axis. This problem covers key concepts in geometry, trigonometry, and algebra, including the tangent function and point-slope form.

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