Problem 5 - Calculus an Applied Approach

Question

Find the intercepts and sketch the graph of the plane. $$ 2 x-y+3 z=4 $$

Step-by-Step Solution

Answer

The x-intercept is 2, the y-intercept is -4, and the z-intercept is \(4/3\). So, the plane passes through the points (2, 0, 0), (0, -4, 0), and (0, 0, 4/3).

Step 1: Find the x-intercept

To find the x-intercept, set the other variables (y and z) equal to zero in the equation \(2x - y + 3z = 4\). This simplifies to \(2x = 4\). Solving for x evaluates to \(x = 2\).

Step 2: Find the y-intercept

Likewise, the y-intercept is found by setting x and z equal to zero. The equation becomes \(-y = 4\), thereby yielding \(y = -4\).

Step 3: Find the z-intercept

Similarly, to find the z-intercept, both x and y are set equal to zero. This simplifies the equation to \(3z = 4\), which solves to \(z = 4/3\).

Step 4: Sketch the graph

After obtaining all the intercepts (2, -4, 4/3), plot these points on a three-dimensional coordinate system. The plane passing through these intercept points is the required solution.

Key Concepts

x-intercept

The x-intercept of a plane in 3D coordinate geometry is where the plane intersects the x-axis. To find the x-intercept, we usually set the other two variables, y and z, to zero and solve for x in the given plane equation.
This removes their influence, making it simple to isolate x.For example, considering the plane equation from our problem: \(2x - y + 3z = 4\), you replace y and z with zero, getting:\(2x = 4\).
This simplifies to \(x = 2\).
So, when the plane crosses the x-axis, it will be at the point (2, 0, 0).

• Set y = 0 and z = 0.
• Reduce the equation to x terms.
• Solve to find the value of x; this gives the x-intercept.

y-intercept

The y-intercept in 3D geometry indicates where a plane intersects the y-axis. To determine this, we set x and z to zero. This approach isolates y in the equation, showcasing its placement along the y-axis.Using the given equation: \(2x - y + 3z = 4\), we plug in x = 0 and z = 0, transforming it to:\(-y = 4\).This simplifies to \(y = -4\).
Hence, the y-intercept is at the point (0, -4, 0) on the y-axis.

• Set x and z to zero.
• Simplify to focus on y.
• Solve to find the y-intercept.

z-intercept

The z-intercept for a plane in three-dimensional coordinates is located where the plane touches the z-axis. To identify this, x and y are each set to zero in the plane equation, highlighting z's role.Taking the equation \(2x - y + 3z = 4\), inputting x = 0 and y = 0 reduces it to:\(3z = 4\).
Solving this gives \(z = \frac{4}{3}\).
Thus, the plane meets the z-axis at (0, 0, \(\frac{4}{3}\)).

• Assign zero to both x and y.
• Simplify to emphasize z-terms.
• Determine the z-intercept through solving.

graph of a plane

Sketching the graph of a plane in 3D geometry involves plotting its intercepts and visualizing the surface extending through these points. This process emphasizes the plane's spatial positioning within a coordinate system.With the points (2, 0, 0), (0, -4, 0), and (0, 0, \(\frac{4}{3}\)), the task is to mark these on a three-dimensional graph properly.Understanding Graph Drawing:

• Start by plotting each intercept point.
• Visualize a flat surface connecting these points.
• This represents a single slice of the infinite plane.

Practically, this drawing step helps in grasping how planes extend beyond their intercepts in all dimensions. It’s like mapping a flat sheet that runs endlessly through the intercepts you've identified.