Question

which of the following could be the equation for the plane graphed above?

\\(x + y - z = 0\\)
\\(x - y + z = 0\\)
\\(-x + y + z = 0\\)
\\(x + y + z = 0\\)

Explanation:

🆕 New Concept Discovered: 3D Plane Visualization
Analyzing intercepts and slopes in three dimensions.

Step 1: Identify the origin and intercepts

Let's look at the center of the coordinate box where \(x = 0\) and \(y = 0\). The plane passes directly through the center of the grid, which corresponds to the origin \((0, 0, 0)\).

All four given options:

1. \(x + y - z = 0\)
2. \(x - y + z = 0\)
3. \(-x + y + z = 0\)
4. \(x + y + z = 0\)

are satisfied by the point \((0, 0, 0)\). We need to look at specific boundary points to determine the correct equation.

Step 2: Analyze boundary points from the graph

Let's observe the values of \(z\) at the corners of the domain where \(x\) and \(y\) range from \(-4\) to \(4\):

• At the far right corner:

Looking at the axes, this corresponds to \(x = -4\) (on the left-to-right axis labeled \(x\) from \(4\) to \(-4\)) and \(y = 4\) (on the back axis labeled \(y\) from \(-4\) to \(4\)).
At this corner, the plane reaches its highest point, where \(z \approx 8\) (well above \(5\)).
Let's test our options with a point like \((x, y) = (-4, 4)\) yielding a positive \(z\):

• For \(x + y - z = 0 \implies z = x + y = -4 + 4 = 0\) (Incorrect, \(z\) is clearly positive here).
• For \(x - y + z = 0 \implies z = y - x = 4 - (-4) = 8\) (Matches perfectly).
• For \(-x + y + z = 0 \implies z = x - y = -4 - 4 = -8\) (Incorrect, \(z\) is positive).
• For \(x + y + z = 0 \implies z = -x - y = -(-4) - 4 = 0\) (Incorrect).

Step 3: Verify with another corner

Let's check the front-left corner of the plane:

• This corresponds to \(x = 4\) and \(y = -4\).
• On the graph, this is the lowest point of the plane, where \(z \approx -8\) (near the bottom of the \(z\)-axis at \(-10\)).
• Let's test our matching equation \(x - y + z = 0 \implies z = y - x\):

$$ z = -4 - 4 = -8 $$

• This perfectly matches the visual representation of the plane.

Answer:

\(x - y + z = 0\)