91Ó°ÊÓ

Equation solving involves finding the values of variables that satisfy a given equation. In 3D geometry, this might mean identifying geometrical shapes defined by the equation. In the exercise, we have \[(y-5)(z-6)=0\]To solve this equation, we look for values of \(y\) and \(z\) that make the product zero. Due to the zero-product property, either - \(y-5=0\) or - \(z-6=0\). Solving these gives: - \(y=5\), and- \(z=6\).Each of these is a separate condition in 3D space. Hence, the solution is not a single point but a series of points satisfying either of these conditions, forming geometrical planes. Understanding this process is critical, as it connects algebraic manipulation to geometric visualization.

Graphing in three-dimensional space involves plotting points defined by \(x\), \(y\), and \(z\) coordinates. This allows us to visualize solutions of equations as geometric shapes. In our exercise, solving for \(y=5\) and \(z=6\) results in planes. A plane \(y=5\) means all values of \(x\) and \(z\) are possible, provided \(y\) is consistently 5. This forms an entire flat surface parallel to the zx-plane at \(y=5\). Similarly, \(z=6\) results in a plane parallel to the xy-plane where \(z\) is always 6. To effectively graph these, you would draw:

• A vertical plane cutting through the yz-axis, parallel to the zx-plane, positioned at \(y=5\).
• A horizontal plane above the xy-plane, parallel and fixed at \(z=6\).

Visualizing these intersecting planes helps to understand the nature of solutions in 3D equations.

Coordinate planes are essentially the flat surfaces in a three-dimensional coordinate system, defined by two axes. Understanding these planes is crucial when working with 3D geometry.

• The xy-plane is the most familiar, lying flat at \(z=0\), spanning horizontally for all \(x\) and \(y\) values.
• The zx-plane lies vertically at \(y=0\), extending infinitely for all \(x\) and \(z\) values.
• The yz-plane is another vertical plane, where \(x=0\), representing all \(y\) and \(z\) values.

In our exercise, the planes \(y=5\) and \(z=6\) serve as extensions or shifts of the zx and xy-planes, respectively. Recognizing how these planes are oriented helps in understanding spatial relationships and how they intersect or complement each other in 3D space.