Question
Does the following equation represent a line in three dimensions or a plane in three dimensions? \( -8 x-5 y+3 z+4=0 \) \( \checkmark \) Select Plane in three dimensions Line in three dimensions
Ask by Adkins Williams. in the United States
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation represents a plane in three dimensions.
Solution
We start with the equation
\[
-8x - 5y + 3z + 4 = 0.
\]
1. In three dimensions, an equation of the form
\[
Ax + By + Cz + D = 0
\]
represents a plane, provided that \(A\), \(B\), and \(C\) are not all zero.
2. In our equation, the coefficients are \(A = -8\), \(B = -5\), and \(C = 3\). Since these are not all zero, the equation defines a plane in three dimensions.
Thus, the equation represents a plane in three dimensions.
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Simplify this solution Beyond the Answer
This equation represents a plane in three dimensions. The general form of a plane's equation can be expressed as \( Ax + By + Cz + D = 0 \), where A, B, and C are constants. In this case, you have coefficients for x, y, and z, which confirms that the given equation describes a plane extending infinitely in all directions defined by those variables. To visualize this, imagine that every point (x, y, z) that satisfies this equation lies on an infinite flat surface. If you were to plot it, you’d see that no matter how far you move within the plane, it stretches out infinitely, unlike a line that would only extend in two dimensions!
