https://r2.erweima.ai/mathbot/image/1725499604951j021fkty.pdf

The document contains exercises on analytic geometry, focusing on equations of lines. Here’s an outline of the key problems from the document:

1. **Finding equations of lines**:
   - Given a point $ P(x_1, y_1) $ and a slope $ m $, find the equation of the line.
     - Examples include $ P(1, 5), m = 2 $, and others with different points and slopes.
   - Given two points $ P(x_1, y_1) $ and $ Q(x_2, y_2) $, find the equation of the line passing through these points.
     - Example: $ P(4, 8) $ and $ Q(0, -5) $.

2. **Finding the equation of a line with specific conditions**:
   - A line with slope $ -3 $ and a y-intercept of $ -2 $.
   - A line passing through a point $ A(7,8) $ and parallel to another line determined by two points $ C(-2, 2) $ and $ D(3, -4) $.

3. **Advanced problems**:
   - Finding the value of $ k $ so that a line is parallel or perpendicular to another given line.
   - Determining the equation of a reflected ray of light based on the intersection with mirrors described by linear equations.

4. **Depreciation problem**:
   - Describing the linear depreciation of books over time from an initial value of $1500 to $0 in 10 years.

Would you like help solving one or more of these problems? Here are five related questions to explore further:

1. How do you find the equation of a line given a point and a slope?
2. How do you calculate the equation of a line passing through two points?
3. What is the method to find the equation of a line parallel to another?
4. How can you find the intersection of two lines using their equations?
5. How is the concept of linear depreciation modeled mathematically?

**Tip**: For lines, the slope-intercept form $ y = mx + b $ is particularly useful when you know the slope and the y-intercept.

The document contains exercises on analytic geometry, focusing on equations of lines. Here’s an outline of the key problems from the document:

1. Finding equations of lines:
   - Given a point P(x_1, y_1) and a slope m, find the equation of the line. Examples include P(1, 5), m = 2.
   - Given two points P(x_1, y_1) and Q(x_2, y_2), find the equation of the line passing through these points.
   - Example: P(4, 8) and Q(0, -5).
2. Finding the equation of a line with specific conditions: A line with slope -3 and a y-intercept of -2.
3. Advanced problems: Finding the value of k so that a line is parallel or perpendicular to another given line. Determining the equation of a reflected ray of light based on the intersection with mirrors described by linear equations.
4. Depreciation problem: Describing the linear depreciation of books over time from an initial value of $1500 to $0 in 10 years.

Learn to solve various problems in analytic geometry, including finding equations of lines given a point and slope, determining parallel and perpendicular lines, and solving depreciation problems over time. Suitable for students in Grades 9-12.

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