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Chapter 10: Problem 19

In Exercises 19 and \(20,\) use the \(x, y,\) and \(z\) intercepts to sketch the plane for each equation. $$x+2 y+z=6$$

Short Answer

Expert verified
The plane intersects the x-axis at (6,0,0), the y-axis at (0,3,0), and the z-axis at (0,0,6).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set both \( y = 0 \) and \( z = 0 \). Substitute these values into the equation \( x + 2y + z = 6 \).\[ x + 2(0) + 0 = 6\]This simplifies to \( x = 6 \). Hence, the x-intercept is at \((6, 0, 0)\).
02

Find the y-intercept

To find the y-intercept, set both \( x = 0 \) and \( z = 0 \). Substitute these values into the equation \( x + 2y + z = 6 \).\[ 0 + 2y + 0 = 6\]This simplifies to \( 2y = 6 \), or \( y = 3 \). Hence, the y-intercept is at \((0, 3, 0)\).
03

Find the z-intercept

To find the z-intercept, set both \( x = 0 \) and \( y = 0 \). Substitute these values into the equation \( x + 2y + z = 6 \).\[ 0 + 0 + z = 6\]This simplifies to \( z = 6 \). Hence, the z-intercept is at \((0, 0, 6)\).
04

Sketch the Plane

Using the intercepts found in Steps 1-3, plot the points \((6, 0, 0)\), \((0, 3, 0)\), and \((0, 0, 6)\) on a 3D coordinate system. Draw a plane that passes through these three intercepts to represent the equation \( x + 2y + z = 6 \). This plane will intersect the x-axis at \((6, 0, 0)\), the y-axis at \((0, 3, 0)\), and the z-axis at \((0, 0, 6)\).

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Key Concepts

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

x-intercept
An intercept is where a graph intersects an axis in a coordinate system. Specifically, the **x-intercept** is the point where the graph meets the x-axis, meaning the values of the other two variables or axes (y and z) are zero.
  • To find the x-intercept in a plane given by an equation like \(x + 2y + z = 6\), you simply set both \(y = 0\) and \(z = 0\) in the equation.
  • This results in \(x = 6\), revealing that the x-intercept is \((6, 0, 0)\).
Once you have substituted these values, you solve for \(x\) by simplifying the equation. Intercepts provide critical points or coordinates that help in graphically representing a plane in a 3D coordinate system. These intersection points are crucial for sketching and visualizing the plane effectively.
y-intercept
The **y-intercept** is where the graph meets the y-axis within a 3D coordinate system. Here, the values of the other two axes (x and z) are zero, which simplifies the calculation of the intercept.To find the y-intercept for the plane given by \(x + 2y + z = 6\):
  • Set \(x = 0\) and \(z = 0\) in the equation. This makes the equation \(0 + 2y + 0 = 6\).
  • Solve for \(y\). In this case, \(2y = 6\) which simplifies to \(y = 3\).
Thus, the y-intercept is at \((0, 3, 0)\). Determining intercepts allows us to understand where a plane intersects the axes of a 3D coordinate system, making it easier to draw or visualize the orientation and position of the plane in space.
z-intercept
The **z-intercept** plays an important role in understanding where a plane meets the z-axis in a 3D coordinate system. For finding this intercept, it is necessary to set the other two coordinates, x and y, to zero:
  • Set \(x = 0\) and \(y = 0\) in the given equation \(x + 2y + z = 6\).
  • This simplifies the equation to \(0 + 0 + z = 6\).
  • Thus, \(z = 6\) gives us the z-intercept at \((0, 0, 6)\).
Understanding intercepts is crucial for positioning the plane into a 3D space accurately. The z-intercept helps in completing the spatial visualization of the plane along with x- and y-intercepts.
plane equations
Plane equations define flat surfaces in a 3D space and are generally expressed in the form \(ax + by + cz = d\). This type of equation helps in identifying the relationship between the three variables x, y, and z that define any point on the plane. To examine the equation \(x + 2y + z = 6\), consider the following:
  • The coefficients \(a, b,\) and \(c\) here correspond to \(1, 2,\) and \(1\) respectively.
  • Knowing how to identify the intercepts, as discussed earlier, allows us to understand and plot where the plane crosses each axis.
A plane can be uniquely determined by finding at least three non-collinear points (like intercepts), allowing it to be sketched accurately in a 3D coordinate system. Understanding plane equations are essential for geometry, physics, and various engineering applications, as they help in modeling real-world phenomena and solving spatial problems.

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Most popular questions from this chapter

Use algebra to find the point of intersection of the two lines whose equations are provided. Use Example 7 as a guide. \(2 x+y=11\) and \(3 x+2 y=16\)

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