Problem 53 - Precalculus with Calculus Previews

Question

Graph the plane whose equation is given. $$ 5 x+2 y+z=10 $$

Step-by-Step Solution

Answer

Plot intercepts: (2,0,0), (0,5,0), (0,0,10); connect to visualize the plane.

Step 1: Understand the plane equation

The equation of the plane is given as \(5x + 2y + z = 10\). This is a linear equation involving three variables, \(x\), \(y\), and \(z\). The plane represents all points \((x, y, z)\) that satisfy this equation.

Step 2: Find intercepts

To graph a plane, it's helpful to find the intercepts where the plane intersects the axes. Start with the \(x\)-intercept by setting \(y = 0\) and \(z = 0\):\[5x + 2(0) + 0 = 10 \implies 5x = 10 \implies x = 2\]Thus, the \(x\)-intercept is \((2, 0, 0)\).Next, find the \(y\)-intercept by setting \(x = 0\) and \(z = 0\):\[5(0) + 2y + 0 = 10 \implies 2y = 10 \implies y = 5\]The \(y\)-intercept is \((0, 5, 0)\).Finally, find the \(z\)-intercept by setting \(x = 0\) and \(y = 0\):\[5(0) + 2(0) + z = 10 \implies z = 10\]The \(z\)-intercept is \((0, 0, 10)\).

Step 3: Plot the intercepts

Plot the intercepts \((2, 0, 0)\), \((0, 5, 0)\), and \((0, 0, 10)\) on a 3D coordinate system. These points represent where the plane intersects the \(x\)-axis, \(y\)-axis, and \(z\)-axis, respectively.

Step 4: Draw the plane

Using the intercepts as reference points, connect the dots in a way that forms a triangle in the 3-dimensional space. This triangle helps visualize the plane within the space. The entire surface extending infinitely in all directions from this triangle represents the plane described by the equation \(5x + 2y + z = 10\).

Key Concepts

3D Coordinate Systems

In geometry, a 3D coordinate system is pivotal for describing and understanding spaces with three dimensions. It extends the basic idea of the 2D coordinate system with an additional dimension, allowing us to describe any point in space using three numbers. Typically, these numbers are expressed as

• The x-coordinate
• The y-coordinate
• The z-coordinate

These values relate to the horizontal, vertical, and depth axes, respectively. This system is crucial for graphing points, lines, and planes in 3D space, offering a distinct advantage over 2D systems by adding the ability to discuss spatial volume and position.

When graphing in a 3D coordinate space, each point \((x, y, z)\) represents a location determined by the distances along the three perpendicular axes from the origin \((0, 0, 0)\) . Here, the concept of depth comes into play, distinguishing 3D coordinates from 2D points.

Linear Equations in Three Variables

Linear equations in three variables, such as \(5x + 2y + z = 10\) , form the basis for describing planes in a three-dimensional space. These equations include three variables:

• \(x\)
• \(y\)
• \(z\)

representing all possible points that satisfy the condition stated in the equation.

Essentially, such equations depict flat surfaces extending infinitely in the 3D space. Each plane divides the space into two halves, with all solutions \((x, y, z)\) falling on this surface.

When a specific point satisfies a linear equation, it means that substituting that point's coordinates into the equation verifies its balance. This balance is crucial in visualizing how points in 3D are aligned on the same flat surface. Understanding these planes allows for robust spatial reasoning, particularly crucial in fields like engineering and physics, where three-dimensional modeling is a necessary part of problem solving.

Intercepts in 3D Geometry

Intercepts in 3D geometry play a key role in graphing planes and visualizing where a plane intersects specific axes. Finding intercepts is an important first step when graphing planes from a linear equation in three variables.

To find an intercept:

• Set two variables equal to zero, keeping the third variable active.
• Solve the equation to find the value for the remaining variable, determining one intercept at a time.

For example, if we consider the x-intercept of the equation \(5x + 2y + z = 10\), we set \(y = 0\) and \(z = 0\), leading to \(5x = 10\), hence \(x = 2\). The intercept points are \((2, 0, 0)\), \((0, 5, 0)\), and \((0, 0, 10)\). These points are crucial as they allow us to plot where the plane cuts through the respective axes in the 3D space, making it easier to visualize the plane's position and orientation.