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Let's go through each problem step-by-step.

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### Problem 7:
1. **Find the intersection point of the two lines:**
   - Given lines:
     \[
     x + 4y = -3
     \]
     \[
     x + y = 0
     \]
   - Solve these two equations simultaneously to find their intersection point.

2. **Determine the slope of the line passing through the points (2, -1) and (4, 6):**
   - Slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
     \[
     m = \frac{y_2 - y_1}{x_2 - x_1}
     \]

3. **Write the equation of the line passing through the intersection point (from Step 1) and parallel to the slope found in Step 2.**

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### Problem 8:
1. **Determine the slope of the line passing through the points (4, 1) and (-2, 2):**
   - Use the slope formula again to find this slope.

2. **Find the slope of the line perpendicular to the line from Step 1:**
   - For two lines to be perpendicular, their slopes are negative reciprocals of each other. So if the slope is \(m\), the perpendicular slope will be \(-\frac{1}{m}\).

3. **Write the equation of the line passing through the point (2, -4) with the perpendicular slope found in Step 2.**

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### Problem 9:
1. **Find the intersection point of the two lines:**
   - Given lines:
     \[
     x + 2y = 3
     \]
     \[
     3x + y = 4
     \]
   - Solve these two equations simultaneously to find their intersection point.

2. **Determine the slope of the line passing through the points (1, 1) and (4, 6):**
   - Use the slope formula.

3. **Find the slope of the line perpendicular to the line from Step 2:**
   - Use the negative reciprocal to find the perpendicular slope.

4. **Write the equation of the line passing through the intersection point (from Step 1) with the perpendicular slope found in Step 3.**

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Would you like detailed solutions for each part, or is there a specific one you would like to explore further? 

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### Related Questions:
1. How do you solve simultaneous linear equations?
2. What is the slope formula, and why does it work?
3. How do you determine if two lines are parallel or perpendicular?
4. What is the general form of a line equation?
5. How do transformations affect the position and slope of a line?

### Tip:
When solving for an intersection point of two lines, use substitution or elimination to simplify the calculations effectively.

Find and graph the equation of the straight line through specific points, parallel or perpendicular to given lines.

Learn to find and graph equations of straight lines passing through given points, parallel or perpendicular to specified lines. This includes solving simultaneous linear equations, using slope formulas, and understanding conditions for parallel and perpendicular lines. Ideal for students in Grades 10-12.

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