What are the intercepts made by the plane 2x+3y+5z+7=0 on the coordinate planes

To find the intercepts of the plane \( 2x + 3y + 5z + 7 = 0 \) with the coordinate planes, we can evaluate the equation by setting two variables equal to zero at a time.

1. **x-intercept**:
   - Set \( y = 0 \) and \( z = 0 \).
   \[
   2x + 7 = 0 \implies x = -\frac{7}{2}
   \]
   Thus, the x-intercept is \( \left( -\frac{7}{2}, 0, 0 \right) \).

2. **y-intercept**:
   - Set \( x = 0 \) and \( z = 0 \).
   \[
   3y + 7 = 0 \implies y = -\frac{7}{3}
   \]
   Thus, the y-intercept is \( \left( 0, -\frac{7}{3}, 0 \right) \).

3. **z-intercept**:
   - Set \( x = 0 \) and \( y = 0 \).
   \[
   5z + 7 = 0 \implies z = -\frac{7}{5}
   \]
   Thus, the z-intercept is \( \left( 0, 0, -\frac{7}{5} \right) \).

### Summary:
- **x-intercept**: \( \left( -\frac{7}{2}, 0, 0 \right) \)
- **y-intercept**: \( \left( 0, -\frac{7}{3}, 0 \right) \)
- **z-intercept**: \( \left( 0, 0, -\frac{7}{5} \right) \)

Would you like more details or have any questions on this?

Here are 5 questions to deepen understanding:
1. How would the intercepts change if the equation of the plane was \( 2x + 3y + 5z - 7 = 0 \)?
2. What is the geometric significance of intercepts on a plane?
3. How can the general equation of a plane be derived from three intercept points?
4. What happens if the constant term in the plane equation is zero?
5. Can the intercepts help in visualizing the orientation of the plane in 3D space?

**Tip:** Always substitute other variables to zero to easily find the intercepts in multi-variable equations.

What are the intercepts made by the plane 2x+3y+5z+7=0 on the coordinate planes?

Learn how to find the x-, y-, and z-intercepts of the plane 2x + 3y + 5z + 7 = 0 using the intercept theorem. This problem involves key concepts of analytic geometry and intercepts in 3D space. Suitable for students in Grades 10-12.

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