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A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in many forms, but their core feature is that they graph as straight lines.
For instance, the equation given in this exercise, \(x + 2y + 3z = 5\), is a classic example of a linear equation.
This specific equation involves three variables, making it a plane in three-dimensional space, where each variable adds another dimension to the geometrical shape it describes. In equations like this, each term \(x, y, z\) can be adjusted to shift the plane in various directions in space.
This can also be seen as equating weighted sums, where each weight (multiplier of a variable) essentially changes the slope or direction of the plane.

Three-dimensional space, often referred to as 3D space, is a geometric setting in which three values are required to determine the position of an element (point). It is a model of the physical space we live in, encompassing width, depth, and height.
In the context of this exercise, the equation \(x + 2y + 3z = 5\) represents a plane within this space.

• Each of the variables \(x, y, z\) corresponds to one of the dimensions.
• When they're combined into an equation form, they define a flat, two-dimensional surface inside the broader 3D space.

On this plane, any point characterized by \(x, y, z\) will satisfy the equation, meaning all solutions to the equation lie on this plane, illustrating part of what makes three-dimensional mathematics so practical for visualizing and solving spatial problems.

Function representation involves expressing one variable in terms of others, assisting in understanding relationships between variables.
In this case, the given equation \(x + 2y + 3z = 5\) can be changed to a function format by solving for \(z\) in terms of \(x\) and \(y\).
This is achieved by rearranging the equation to: \ z = \frac{5 - x - 2y}{3} \, which shows how \(z\) changes with respect to \(x\) and \(y\).

• This approach converts the level surface (plane) into a function \(f(x, y)\).
• It provides a way to "graph" the surface in a more understandable manner, indicating how changes in \(x\) and \(y\) directly affect \(z\).

By isolating \(z\), this function representation simplifies comprehension and visual representation, reflecting a plane in 3D with clarity.