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Chapter 13: Problem 23

Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. $$x+3 y-5 z-30=0$$

Short Answer

Expert verified
The intersection points are (30, 0, 0) on the x-axis, (0, 10, 0) on the y-axis, and (0, 0, -6) on the z-axis. 2. What is the equation of the line where the plane intersects the xy-plane? The equation of the line is x + 3y - 30 = 0. 3. What is the equation of the line where the plane intersects the yz-plane? The equation of the line is 3y - 5z - 30 = 0. 4. What is the equation of the line where the plane intersects the xz-plane? The equation of the line is x - 5z - 30 = 0. 5. How would you sketch the graph of the plane? To sketch the graph of the plane, plot the intersection points (30, 0, 0), (0, 10, 0), and (0, 0, -6) with the coordinate axes, and connect them to form a triangle representing the plane.

Step by step solution

01

Find the intersection points with the coordinate axes

We will begin by finding the intersection points of the plane equation \(x + 3y - 5z - 30 = 0\) with the \(x\)-, \(y\)-, and \(z\)-axes. For the \(x\)-axis, let \(y = 0\) and \(z = 0\). We get: $$x + 3(0) - 5(0) - 30 = 0$$ $$x = 30$$ So, the intersection point with the \(x\)-axis is \((30, 0, 0)\). For the \(y\)-axis, let \(x = 0\) and \(z = 0\). We get: $$0 + 3y - 5(0) - 30 = 0$$ $$3y = 30$$ $$y = 10$$ So, the intersection point with the \(y\)-axis is \((0, 10, 0)\). For the \(z\)-axis, let \(x = 0\) and \(y = 0\). We get: $$0 + 3(0) - 5z - 30 = 0$$ $$-5z = 30$$ $$z = -6$$ So, the intersection point with the \(z\)-axis is \((0, 0, -6)\).
02

Obtain the equations of the lines where the plane intersects the coordinate planes

Now we will find the equations of the lines where this given plane intersects each of the coordinate planes. 1. Intersection with the \(xy\)-plane (\(z = 0\)): The given plane equation becomes: $$x + 3y -30 = 0$$ This is the line equation where the plane intersects the \(xy\)-plane. 2. Intersection with the \(yz\)-plane (\(x = 0\)): The given plane equation becomes: $$3y -5z - 30 = 0$$ This is the line equation where the plane intersects the \(yz\)-plane. 3. Intersection with the \(xz\)-plane (\(y = 0\)): The given plane equation becomes: $$x - 5z - 30 = 0$$ This is the line equation where the plane intersects the \(xz\)-plane.
03

Sketch a graph of the plane

To sketch the graph of the plane, plot the intersection points with the coordinate axes and connect them to form a triangle. 1. Intersection point with the \(x\)-axis: \((30, 0, 0)\) 2. Intersection point with the \(y\)-axis: \((0, 10, 0)\) 3. Intersection point with the \(z\)-axis: \((0, 0, -6)\) Draw the lines connecting these points to form a triangle, the graph of the given plane.

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