Final answer:
To find the intersection point between the given line and plane, we express x, y, and z in terms of a single variable, solve for that variable, and then calculate y and z accordingly. The point of intersection is (-9.5, -0.5, 4.75).
Step-by-step explanation:
The question asks to find the point of intersection between a line given by the equations x = y - 9 = -2z and the plane -y + 2z = 1. To solve this, we need to express x, y, and z in terms of a single variable. From x = y - 9, we get y = x + 9. Substituting y in -y + 2z = 1, we have -(x + 9) + 2z = 1, which simplifies to -x - 9 + 2z = 1. Adding 9 to both sides and adding x to both sides gives 2z = x + 10. Using the third equation given for the line, -2z = x - 9, we can equate the right-hand side of the two expressions for z, resulting in x + 10 = -(x - 9). By solving this equation for x, we find x = -9.5. Substitute -9.5 for x into y = x + 9 to get y = -0.5 and into -2z = x - 9 to get z = 4.75. Therefore, the point of intersection is (-9.5, -0.5, 4.75).