Question

Label any intercepts and sketch a graph of the plane. $$ y+z=5 $$

Short answer

The plane intersects the y-axis at \((0,5,0)\) and the z-axis at \((0,0,5)\). Any x-values are possible for this plane.

Step-by-step solution

Find the intercepts

The intercepts of a plane can be found by setting two of the variables to zero and solving for the remaining variable. For the x-intercept, let \(y = z = 0\). Because the equation does not contain \(x\), it means \(x\) could have any value and we simply focus on \(y\) and \(z\). Hence the x-intercept is not defined in a conventional sense. Now, for the y-intercept, let \(x = z = 0\) and solve for \(y\), and for the z-intercept, let \(x = y = 0\) and solve for \(z\).

Solve the equations

For the y-intercept: \(x = z = 0 \rightarrow y = 5\). Hence the graph of the plane will intersect the y-axis at \((0,5,0)\). For the z-intercept: \(x = y = 0 \rightarrow z = 5\). Hence, the graph of the plane will intersect the z-axis at \((0,0,5)\). Thus the intercepts are \((0,5,0)\) and \((0,0,5)\)

Sketch the graph

To sketch the graph, plot these intercept points onto a set of 3D axes. Then draw a line connecting the intercepts on the y and z axis. Finally, draw horizontal lines from these points parallel to the x-axis, thereby creating a plane.

Key concepts

Understanding Coordinate Plane Intercepts

Visualizing the points at which a plane crosses the axes in a three-dimensional space can be crucial in understanding its orientation and position. In three dimensions, a plane can potentially intercept each of the x, y, and z axes.

When a plane is described by an equation such as \( y + z = 5 \), identifying its intercepts involves setting two of the three variables to zero and solving for the remaining one. This step determines the points where the plane touches the axes, known as the 'intercepts'. However, if the plane's equation lacks a variable, like the x-coordinate in this case, it indicates that the plane runs parallel to that axis and doesn't intercept it at a fixed point, but rather at any x-value.

To clarify with an example, let's consider the equation from the problem at hand. If you need to find the y-intercept, set the other variables to zero (\( x = z = 0 \)) and solve for y, yielding the point \( (0,5,0) \). Similarly, setting \( x = y = 0 \), and solving for z gives the z-intercept at \( (0,0,5) \). The axis that is missing from the equation doesn't have a specific intercept point because the plane never crosses it at a fixed location.

Mastering 3D Graphing

Graphing in three dimensions is an extension of the familiar 2D graphing, introducing an additional axis to represent depth. In 3D graphing, each point is defined by three coordinates \((x, y, z)\), indicating its position relative to three mutually perpendicular planes.

To graph a plane, you first plot its intercepts on the coordinate axes. These serve as anchor points from which the plane extends. In the absence of a value for one coordinate (like the x-intercept in our exercise), the plane extends infinitely along that axis.

Practical Tips for 3D Graphing

• Start by sketching the three axes, labeling them x, y, and z appropriately.
• Mark the intercepts on the respective axes.
• Connect these points and visualize how the plane extends between and beyond them.
• Use dashed lines to represent parts of the plane that are hidden from view.
• When possible, include additional points by solving for one variable while assigning particular values to the others to confirm the orientation of the plane.

Understanding the spatial relationship in 3D graphing is essential to accurately interpreting and solving problems in subjects such as physics, engineering, and multidimensional calculus.

Solving Equations in Three Variables

Solving equations in three variables often involves finding solutions that satisfy all three variables simultaneously. Typically, you deal with systems of equations, but with a single equation like \( y + z = 5 \) you’re identifying specific points or a set of points that lie on the plane it represents.

The strategy for solving such an equation is straightforward: selectively set variables to zero to find intercepts, or assign values to two variables and solve for the third to plot additional points on the plane.

Strategic Steps to Solve Equations:

• Identify what you're solving for – in this case, where the plane intersects each axis.
• Set the appropriate variables to zero to find each intercept.
• Use the intercepts to understand the orientation of the plane in the coordinate space.
• To find additional points on the plane, select values for two variables and solve for the third.

The ability to solve such equations is vital for creating graphs and models in higher mathematics, physics, and many engineering fields, aiding in visualizing complex spatial relationships and behaviors.