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Chapter 29: Problem 6

Sketch the graphs of the given equations in the rectangular coordinate system in three dimensions. $$2 y+3 z=6$$

Short Answer

Expert verified
The equation \(2y + 3z = 6\) represents a plane parallel to the \(x\)-axis with intercepts at \((0,3,0)\) and \((0,0,2)\).

Step by step solution

01

Identify the Given Equation Type

The given equation is a linear equation in three variables, specifically involving the variables \(y\) and \(z\), with \(x\) implicitly present as a variable. In three-dimensional space, such equations represent planes.
02

Rewriting the Equation in Plane Form

Rewrite the given equation \(2y + 3z = 6\) in a way that emphasizes it as a plane by expressing it as \(2y + 3z = 6\) without ties to a specific variable. This is already in a suitable form for 3D plotting and shows that the plane is not dependent on \(x\).
03

Finding Intercepts on Axes

To sketch the plane, find intercepts on the \(y\) and \(z\) axes.- **\(y\)-axis intercept**: Set \(z = 0\). Solving \(2y = 6\) gives \(y = 3\), so point \((0, 3, 0)\).- **\(z\)-axis intercept**: Set \(y = 0\). Solving \(3z = 6\) gives \(z = 2\), so point \((0, 0, 2)\).
04

Sketching the Plane

Plot the intercepts \((0, 3, 0)\) and \((0, 0, 2)\) in the 3D coordinate system. Since the equation does not contain the variable \(x\), the plane is parallel to the \(x\)-axis, sweeping through all values of \(x\) for these intercepts. Imagine extending lines parallel to the \(x\)-axis from each intercept and connecting them to form a flat surface, which would be the graph of the plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are mathematical statements that establish a constant relationship between variables. In three dimensions, such equations often incorporate two or three variables. They take the form of \( ax + by + cz = d \), where \(a\), \(b\), and \(c\) are coefficients, and \(d\) is a constant term.
In this scenario, the equation \(2y + 3z = 6\) is a linear equation involving \(y\) and \(z\). Even though \(x\) is not explicitly mentioned, it is implicitly present, defining a plane within a 3D space.
These equations precisely determine the orientation and position of geometrical objects like lines or planes in space, making them a crucial component of 3D coordinate geometry.
Intercepts
Intercepts are essential points where a line or plane intersects one of the coordinate axes. In the context of 3D coordinate systems, intercepts help visualize and construct the geometric object being described, such as a plane.
For the equation \(2y + 3z = 6\), identifying intercepts is crucial for accurately sketching the plane.
  • On the \(y\)-axis: To find the \(y\)-intercept, set \(z = 0\). Solving \(2y = 6\) gives \(y = 3\), which corresponds to the intercept point \((0, 3, 0)\).

  • On the \(z\)-axis: Here, set \(y = 0\). Solving \(3z = 6\) results in \(z = 2\), leading to the \(z\)-intercept point \((0, 0, 2)\).
These intercept points are crucial landmarks for constructing the rest of the plane.
Planes
In three-dimensional geometry, planes are flat, two-dimensional surfaces extending infinitely in all directions within their boundary constraints. They are depicted by linear equations involving at least two variables.
For the equation \(2y + 3z = 6\), the plane does not depend on \(x\), making it parallel to the \(x\)-axis. This characteristic means the plane is uniform and stretches indefinitely along the \(x\)-direction.
Understanding planes in 3D helps in visualizing how surfaces intersect, overlap, and relate to other spatial figures. Recognizing how planes relate to linear equations aids in constructing, interpreting, and manipulating complex spatial models.
Graph Sketching
Graph sketching, especially in a 3D coordinate system, involves plotting equations to visually represent their geometric properties. For linear equations like \(2y + 3z = 6\), graph sketching helps in constructing the visual shape of the plane.
To sketch this equation in 3D, first, plot the intercepts: \((0, 3, 0)\) and \((0, 0, 2)\). From these points, envision lines extending parallel to the \(x\)-axis. Connecting these extensions across various \(x\) values forms the complete plane.
  • Label axes and intercepts clearly to avoid confusion.
  • Ensure lines representing the plane respect the parallel nature to the \(x\)-axis.
Graph sketching provides a tangible way to interpret mathematical relationships within and across various geometrical spaces.

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