Find the intersection between the plane 
4x−y+5z=5
 and the line through the points 
(0,0,1)
 and 
(2,1,0)
.

To find the intersection between the plane \( 4x - y + 5z = 5 \) and the line passing through the points \((0, 0, 1)\) and \((2, 1, 0)\), we can proceed as follows:

1. **Find the parametric equation of the line:**

The line through the points \((0, 0, 1)\) and \((2, 1, 0)\) can be parameterized as:
\[
\mathbf{r}(t) = (1-t) (0, 0, 1) + t (2, 1, 0)
\]
Simplifying this, we get:
\[
\mathbf{r}(t) = (2t, t, 1-t)
\]

So, the parametric equations for the line are:
\[
x = 2t, \quad y = t, \quad z = 1 - t
\]

2. **Substitute the parametric equations into the plane equation:**

The plane equation is:
\[
4x - y + 5z = 5
\]

Substituting \( x = 2t \), \( y = t \), and \( z = 1 - t \) into the plane equation, we get:
\[
4(2t) - t + 5(1 - t) = 5
\]

Simplify this equation:
\[
8t - t + 5 - 5t = 5
\]
\[
2t + 5 = 5
\]
\[
2t = 0
\]
\[
t = 0
\]

3. **Find the coordinates of the intersection point:**

Substitute \( t = 0 \) back into the parametric equations:
\[
x = 2(0) = 0, \quad y = 0, \quad z = 1 - 0 = 1
\]

So, the intersection point is \((0, 0, 1)\).

Therefore, the intersection between the plane \( 4x - y + 5z = 5 \) and the line through the points \((0, 0, 1)\) and \((2, 1, 0)\) is at the point \((0, 0, 1)\).

Would you like more details or have any questions?

Here are some additional questions you might find interesting:
1. How do you find the parametric equation of a line given two points?
2. What is the general form of the equation of a plane?
3. How can you determine if a point lies on a given plane?
4. How do you find the intersection of two planes?
5. What are the different methods for finding the intersection of a line and a plane?

**Tip:** When finding intersections between geometric objects like lines and planes, converting everything into a consistent form (such as parametric or Cartesian equations) can simplify the problem significantly.

Learn how to find the intersection between the plane 4x−y+5z=5 and the line passing through the points (0,0,1) and (2,1,0). Follow a detailed step-by-step solution including the parametric equation of the line and substitution into the plane equation. Suitable for advanced high school students and beyond.

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