equation x+3y+2z=6

(6,2,3)

how would you sketch the plane on a set of axes? is it supposed to look like a triangle? becuse my paper says it makes a trangular pyramid. i don't get it

draw dot for (0,0,0)

draw z axis up from the dot
draw x axis toward lower right from dot
draw y axis toward upper right from dot
NOW
when x and y are 0, z = 3, mark spot (0,0,3)
when x and z are 0, y = 2, mark spot (0,2,0)
when y and z are 0, x = 6, mark spot (6,0,0)
Note the three spots make a pyramid.
when x and

To sketch the plane represented by the equation x + 3y + 2z = 6 on a set of axes, we usually need to find three points on the plane. One way to do this is to set two of the variables to arbitrary values and solve for the third variable.

Let's set y = 0 and z = 0. By substituting these values into the equation, we get x + 0 + 0 = 6, which simplifies to x = 6. This gives us the point (6, 0, 0).

Now, let's set x = 0 and z = 0. Substituting these values, we get 0 + 3y + 0 = 6, which simplifies to 3y = 6 or y = 2. This gives us the point (0, 2, 0).

Finally, let's set x = 0 and y = 0. Substituting these values, we get 0 + 0 + 2z = 6, which simplifies to 2z = 6 or z = 3. This gives us the point (0, 0, 3).

Now we have three points: (6, 0, 0), (0, 2, 0), and (0, 0, 3). Plotting these points on a set of axes will give you three non-collinear points on the plane.

To determine whether the plane is triangular or not, we need to check if the three points lie on a straight line. If they do, the plane is not triangular. If they don't, the plane is triangular.

Calculating the equation of the line passing through points (6, 0, 0), (0, 2, 0), and (0, 0, 3) can help us determine this. If the equation of the line involves only two variables, then the points lie on a line, and the plane is not triangular.

In this case, we can calculate the equation of the line by finding the direction ratios of two vectors in the plane, which can be obtained by subtracting one point from the other two points:

Vector 1 = (0, 2, 0) - (6, 0, 0) = (-6, 2, 0)
Vector 2 = (0, 0, 3) - (6, 0, 0) = (-6, 0, 3)

Now, we can check if the direction ratios of these two vectors are proportional or not. If they are proportional, the points lie on a line, and the plane is not triangular. If they are not proportional, then the points do not lie on a line, and the plane is triangular.

Comparing the direction ratios, (-6, 2, 0) and (-6, 0, 3), we can say that they are not proportional to each other. Therefore, the points (6, 0, 0), (0, 2, 0), and (0, 0, 3) are not collinear, and the plane is triangular.

In conclusion, the plane represented by the equation x + 3y + 2z = 6 can be sketched on a set of axes by plotting the three non-collinear points (6, 0, 0), (0, 2, 0), and (0, 0, 3), forming a triangular shape.