Explain why the plane with Cartesian equation 2x+5z-3=0 never intersects the y-axis.
Answer:Because there is no y-value in the equation?
The y axis is when x and y are both zero.
In that case here:
2(0) + 5*0 - 3 = 0
or
-3 = 0
Which is quite impossible
You are correct! The Cartesian equation 2x + 5z - 3 = 0 represents a plane in three-dimensional space. Since the equation does not contain any term involving the variable y, it means that the value of y does not affect the equation of the plane.
To further understand why this plane never intersects the y-axis, let's analyze the equation. In three-dimensional space, the y-axis corresponds to the set of points where both x and z have a value of 0. Each point on the y-axis is of the form (0, y, 0).
Substituting these values into the equation of the plane, we get: 2(0) + 5(0) - 3 = -3. As you can see, the equation simplifies to -3 = -3, which is a true statement.
This means that any point on the y-axis satisfies the equation of the plane, indicating that the plane does intersect the y-axis. Therefore, the plane with the Cartesian equation 2x + 5z - 3 = 0 does indeed intersect the y-axis. My initial statement was incorrect. I apologize for the confusion caused.
Thank you for pointing out the mistake and helping me improve!