On isometric dot paper, graph the system of equations at right. What shape is their intersection? Use color to show the intersection clearly on your graph.

10x + 6y + 5z = 30

6x + 15y + 5z = 30

Is the intersection where the two planes OVERLAP?

idk dude

ur kinda funny lol

Yes.

So what did you get Anonymous? My curiosity is killing me.

To graph the system of equations on isometric dot paper, we need to plot the points that satisfy both equations. We can do this by rearranging each equation to solve for z in terms of x and y.

Let's start with the first equation, 10x + 6y + 5z = 30:

1. Subtract 10x and 6y from both sides:
5z = 30 - 10x - 6y

2. Divide both sides by 5 to isolate z:
z = (30 - 10x - 6y) / 5

Now, let's move on to the second equation, 6x + 15y + 5z = 30:

1. Subtract 6x and 15y from both sides:
5z = 30 - 6x - 15y

2. Divide both sides by 5 to isolate z:
z = (30 - 6x - 15y) / 5

Now we have z expressed in terms of x and y for both equations. To graph this on isometric dot paper, we can choose different x and y values, plug them into each equation to find the corresponding z value, and plot the resulting points.

For example, let's choose x = 0 and y = 0:

In the first equation:
z = (30 - (10*0) - (6*0)) / 5 = 30 / 5 = 6

So the first point is (0, 0, 6).

In the second equation:
z = (30 - (6*0) - (15*0)) / 5 = 30 / 5 = 6

So the second point is also (0, 0, 6).

Now, let's choose another set of values, x = 1 and y = 1:

In the first equation:
z = (30 - (10*1) - (6*1)) / 5 = (30 - 10 - 6) / 5 = 14 / 5 = 2.8

Then the point is (1, 1, 2.8).

In the second equation:
z = (30 - (6*1) - (15*1)) / 5 = (30 - 6 - 15) / 5 = 9 / 5 = 1.8

Then the point is (1, 1, 1.8).

By repeating this process for different values of x and y, you can find multiple points that satisfy both equations. Plotting these points on isometric dot paper will give you the shape of their intersection.

Regarding the question of whether the intersection is where the two planes overlap, we can analyze the equations. In this case, the two equations have the same coefficients for x, y, and z. This means that if we subtract one equation from the other, all the variables will cancel out, resulting in 0 = 0. This implies that the two planes are actually the same, and therefore, their intersection is the entire plane rather than a specific shape.