"Write a system of equations in three variables that has a single solution of (-1, -2, -3)"
The first equation that I made was x+y+z = -6, but I'm not sure on how to continue with it.
How would I write a system of equations in three variables that has no solution?
That's a great start. You could continue by creating some more true statements based on the given values;
e.g. 4(-1) - (-2) - (-3) = 1, so 4x-y-z = 1
etc.
To write a system of equations with a single solution of (-1, -2, -3), we need three equations since we have three variables. We can start by using your equation x + y + z = -6 and build upon it.
To create the second equation, we can use the fact that the x-coordinate of the solution is -1. Therefore, we can write an equation where the x-value is explicitly -1. Let's use the equation:
x + 2y - 3z = -1
Now, we have two equations:
1) x + y + z = -6
2) x + 2y - 3z = -1
To create the third equation, we can use the fact that the y-coordinate of the solution is -2. Therefore, we can write another equation where the y-value is explicitly -2. Let's use the equation:
2x + y + 4z = -2
Now, we have the complete system of equations:
1) x + y + z = -6
2) x + 2y - 3z = -1
3) 2x + y + 4z = -2
This system of equations has a single solution of (-1, -2, -3).