Find an equation for the level surface of the function through a given point. x-y + 2z (1,0, -1) x+y-z ... An equation for the level surface passing through the point (1,0, -1) is z =
Find an equation for the level surface of the function through a given point. x-y + 2z (1,0, -1) x+y-z ... An equation for the level surface passing through the point (1,0, -1) is z =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 37E
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![### Finding the Level Surface Equation for a Given Function and Point
Given a function and a specific point, we aim to find the equation for the level surface that passes through this point.
Consider the function:
\[
\frac{x - y + 2z}{x + y - z}
\]
We are given the point \((1, 0, -1)\).
**Task:**
Find the equation for the level surface of the function that passes through the given point.
**Solution Steps:**
1. Substitute the coordinates of the point \((1, 0, -1)\) into the function:
\[
f(1, 0, -1) = \frac{1 - 0 + 2(-1)}{1 + 0 - (-1)} = \frac{1 - 2}{1 + 1} = \frac{-1}{2} = -\frac{1}{2}
\]
2. This value represents the constant \( k \) for the level surface. Thus, the level surface equation is:
\[
\frac{x - y + 2z}{x + y - z} = -\frac{1}{2}
\]
Hence, the equation for the level surface passing through the point \((1, 0, -1)\) is:
\[
\frac{x - y + 2z}{x + y - z} = -\frac{1}{2}
\]
You can substitute \( x \), \( y \), and \( z \) values into this equation to verify that any point on this surface will satisfy the equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a1af84-f79e-495e-9bf6-e155c46aa5b8%2F0ab6636e-4db9-4553-9ad7-e871f48c3bfc%2Fqlo0rf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Level Surface Equation for a Given Function and Point
Given a function and a specific point, we aim to find the equation for the level surface that passes through this point.
Consider the function:
\[
\frac{x - y + 2z}{x + y - z}
\]
We are given the point \((1, 0, -1)\).
**Task:**
Find the equation for the level surface of the function that passes through the given point.
**Solution Steps:**
1. Substitute the coordinates of the point \((1, 0, -1)\) into the function:
\[
f(1, 0, -1) = \frac{1 - 0 + 2(-1)}{1 + 0 - (-1)} = \frac{1 - 2}{1 + 1} = \frac{-1}{2} = -\frac{1}{2}
\]
2. This value represents the constant \( k \) for the level surface. Thus, the level surface equation is:
\[
\frac{x - y + 2z}{x + y - z} = -\frac{1}{2}
\]
Hence, the equation for the level surface passing through the point \((1, 0, -1)\) is:
\[
\frac{x - y + 2z}{x + y - z} = -\frac{1}{2}
\]
You can substitute \( x \), \( y \), and \( z \) values into this equation to verify that any point on this surface will satisfy the equation.
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