To graph a plane, you will need at least two lines contained in the plane. Begin by determining the lines for each plane.
Practice makes perfect
Before we begin, remember that two lines are enough to define a plane. Therefore, we will need two lines for each plane in order to graph them.
2x+3y-z=12 & (I) 2x-4y+z=8 & (II)
Let's graph each plane separately.
Plane (I)
The lines contained in Plane (I) can be found by substituting 0 for one of its variables. The resulting equation represents a line. Let's first find the equation of the line that passes through x= 0.
Now that we know the equation of the line, we can use its intercepts to graph it. For the y-intercept, we will substitute z= 0. Similarly, for the z-intercept, we will substitute y= 0.
y-intercept
z-intercept
Substitute
3y- 0=12
3( 0)-z=12
Calculate
y= 4
z= -12
Point
( 0, 4, 0)
( 0, 0, -12)
Next, we can plot the intercepts on the coordinate space and draw the line through them.
Now we will find the equation of another line on the plane. This time, let's look for a line that passes through y= 0.
To graph the line, we will find its intercept by substituting z= 0 for the x-intercept and x= 0 for the z-intercept.
x-intercept
z-intercept
Substitute
2x- 0=12
2( 0)-z=12
Calculate
x= 6
z= -12
Point
( 6, 0, 0)
( 0, 0, -12)
Now that we know the intercepts, let's graph the second line!
Finally, we have two lines to graph a plane.
Plane (II)
We will graph Plane (II) in the same way as we graphed Plane (I). Let's first find two lines such that one passes through x= 0 and the second through y= 0.
Substitution
Resulting Equation
x=0
2( 0)-4y+z=8
-4y+z=8
y=0
2x-4( 0)+z=8
2x+z=8
Next, we will find the intercepts of each line to graph them.
-4y+z=8
2x+z=8
Intercept
y-intercept
z-intercept
x-intercept
z-intercept
Substitution
-4y+ 0=8
-4( 0)+z=8
2x+ 0=8
2( 0)+z=8
Calculation
y= - 2
z= 8
x= 4
z= 8
Point
( 0, - 2, 0)
( 0, 0, 8)
( 4, 0, 0)
( 0, 0, 8)
Now that we know the intercepts of both lines, we can graph them.
Now we can graph the plane that contains the lines.
Combining the Graphs
In the final step, we will graph the planes in the same coordinate system.
