The equation you provided, $2x + 3y - z = 5$, represents a plane in 3D space. Here's how we can understand the graph of this equation:

1. Equation Type: This is a linear equation in three variables (x, y, and z), so the graph is a plane, not a curve. The general form of a plane in 3D space is $ax + by + cz = d$, where $a$, $b$, and $c$ are constants, and $d$ is a constant as well.

2. Rearranging the Equation: You can express the equation in a form that gives you a better sense of how to plot it: $z = 2x + 3y - 5$ This shows the plane in terms of $z$, which allows you to easily calculate $z$ values given $x$ and $y$ values.

3. Intercepts:

   • X-intercept: Set $y = 0$ and $z = 0$, and solve for $x$: $2x = 5 \quad \Rightarrow \quad x = \frac{5}{2}$
   • Y-intercept: Set $x = 0$ and $z = 0$, and solve for $y$: $3y = 5 \quad \Rightarrow \quad y = \frac{5}{3}$
   • Z-intercept: Set $x = 0$ and $y = 0$, and solve for $z$: $z = -5$

4. Shape: This plane extends infinitely in all directions within the 3D space, cutting through the x-, y-, and z-axes at the points calculated above. The slope and orientation of the plane are determined by the coefficients of $x$ and $y$.

Would you like a visual representation of the plane? I can generate a 3D plot of this equation for you.

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Further Questions:

1. How would you graph other linear equations in 3D space?
2. What happens if we change the constants in the equation, like using $2x + 3y + z = 5$?
3. How can we use the normal vector to describe the orientation of the plane?
4. What is the angle between two planes?
5. How do you interpret the solution to a system of linear equations geometrically?

Tip: When graphing planes, it's helpful to first find the intercepts and plot them. Connecting