2x + 3y + 4z = 12
3-Dimensional Cartesian coordinate system.
x-axis, y-axis, and z-axis.
The z-axis is perpendicular to the xy-plane at the origin.
z
The xy-plane is horizontal.
7
6
The yz-plane is vertical.
5
4
3
2
1
x
7
6
5
4
3
2
1
The xz-plane is vertical.
1 2 3 4 5 6 7
3D graphs will appear to have
the correct proportions if the xy axis, coming “out” from the
page makes a 45˚ angle with
the other 2 axis, and has a
scale of about 0.7 times the
spacing on the other 2 axis.
z
4
-4
-4
4
y
4
x
-4
As you show more of the axis, it becomes harder to “read” the graph,
the negative sections are only drawn when needed.
Ordered triples solve the equation: (x, y, z)
2x + 3y + 4z = 12
If z = 0, then 2x + 3y + 4(0) = 12
z
2x + 3y = 12
If z = 1, then 2x + 3y + 4(1) = 12
2x + 3y = 8
4
If z = 2, then 2x + 3y + 4(2) = 12
2x + 3y = 4
4
4
x
y
The graph is a plane cutting
through all three axis.
All that is important to the graph are the
three green lines.
These traces show where the plane of the
graph hits each Cartesian coordinate planes.
The xy-trace is located on the xy-plane. It can be found when z = 0.
The yz-trace is located on the yz-plane. It can be found when x = 0.
The xz-trace is located on the xz-plane. It can be found when y = 0.
You can draw a reasonable picture using just these three traces.
2x + 3y + 4z = 12
If z = 0, 2x + 3y = 12.
If x = 0, 3y + 4z = 12.
If y = 0, 2x + 4z = 12.
If x = 0, y = 4
If y = 0, z = 3
If x = 0, z = 3
If y = 0, x = 6
If z = 0, y = 4
If z = 0, x = 6
The x-intercept is 6. It is the value of x when both y and z equal
zero. To graph the traces, all you really need is to graph the three
intercepts. Then connect them.
2x + 3y + 4z = 12
x-intercept: (6, 0, 0)
y-intercept: (0, 4, 0)
z-intercept: (0, 0, 3)
z
4
yz-trace
xz-trace
y
4
4
x
xy-trace
15x + 10y + 6z = 60
z
4
y
4
4
x
Solve a system with 3 equations, 3 unknowns.
3x + 2y - 4z = 1
5x - 3y + 7z = 28
2x - 4y + 3z = 17
6(3x + 2y - 4z = 1)
4(5x - 3y + 7z = 28)
3(2x - 4y + 3z =17)
18x + 12y - 24z = 6
20x - 12y + 28z = 112
6x - 12y + 9z = 51
18x + 12y - 24z =
38x
6
+ 4z = 118
18x + 12y - 24z = 6
24x
- 15z = 57
Put x into one of the 2variable equations.
8(3) - 5z = 19
24 - 5z = 19
19x + 2z = 59
8x - 5z = 19
5(19x + 2z = 59)
2(8x - 5z = 19)
95x + 10z = 295
16x - 10z = 38
-5z = -5
z=1
Put both x and z into an original equation to find y.
95x + 10z = 295
111x
= 333
x=3
Solve a system with 4 equations, 4 unknowns.
w - 5x + 2y -
z = -18
3w + x - 3y + 2z = 17
4w - 2x + y -2w + 3x -
z = -1
y + 4z = 11