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When attempting to find the x-intercept of a plane in 3D space, represented by the equation 3x + 6y + 2z = 6, we look for the point where the plane crosses the x-axis. This is found by setting the values of the other variables, y and z, to zero.

By substituting y = 0 and z = 0 into our equation, it simplifies to 3x = 6. We then solve for x, which represents the distance from the origin to the point where the plane intersects the x-axis. In this particular equation, the x-intercept is calculated to be x = 2, hence the point of intersection is (2,0,0).

Locating the y-intercept requires us to find where the plane meets the y-axis. This means both x and z must be zero.

Using the equation 3x + 6y + 2z = 6 and setting x = 0 and z = 0, we're left with 6y = 6. Solving for y gives us the value of 1, which represents the y-intercept of the plane. Subsequently, the coordinate of the y-intercept is noted as (0,1,0).

• Set x and z to zero to focus on the y-variable.
• Solve the resulting equation for y.
• Plot this point on the y-axis to represent the y-intercept.

To determine where the plane cuts through the z-axis, known as the z-intercept, we set both x and y to zero in the original equation. This method isolates the z variable.

In the given equation, introducing x = 0 and y = 0 turns the equation into 2z = 6. Solving this, z is found to be 3, therefore the z-intercept is at (0,0,3).

• Isolate z by making x and y zero.
• Solve for z to get the z-intercept.
• The z-intercept point is then plotted on the z-axis.

Understanding this step is crucial in the overall process of graphing the plane.

Sketching a plane within a 3D coordinate system entails plotting the x, y, and z-intercepts and connecting them to form a triangular shape, which acts as part of the plane.

For our equation 3x + 6y + 2z = 6, we have the intercepts (2,0,0), (0,1,0), and (0,0,3). We start by plotting these points on their respective axes. After marking these points, we draw lines to connect them, forming a triangle that is a portion of the plane. This shape helps visualize the orientation and position of the plane. Finally, to represent the actual plane, we extend this triangle infinitely along its breadth and width, making sure that it remains consistent with its intercepts.

Through practice, sketching a 3D graph can become intuitive, aiding in better comprehension of the spatial relationship between planes and the coordinate axes.