Final answer:
To solve the system of equations, we can use the elimination method to eliminate one variable at a time. By subtracting the second and third equations from the first equation, we can simplify the system of equations. Finally, we can solve for x, y, and z and write the solution as an ordered triple.
Step-by-step explanation:
To solve the system of equations:
2x + y + 3z = 20
x + 2y - z = -11
3x + 2z = -3
We can start by using the elimination method to eliminate one variable at a time. We can multiply the second equation by 2, and the third equation by -1:
2x + y + 3z = 20
2x + 4y - 2z = -22
-3x - 2z = 3
Now we can subtract the second equation from the first equation and the third equation from the first equation:
2x + y + 3z - (2x + 4y - 2z) = 20 - (-22)
-3x - 2z - (2x + y + 3z) = 3 - 20
This simplifies to:
-3y + 5z = 42
-5x - y - 5z = -17
From the first equation, we can solve for y:
-3y = -5z + 42
y = (5z - 42)/3
Now we can substitute this expression for y into the second equation:
-5x - (5z - 42)/3 - 5z = -17
Now we can solve this equation for x:
-5x - (5z - 42)/3 - 5z = -17
And finally, we can substitute the values of x and y back into any one of the original equations to solve for z. Once we have the values of x, y, and z, we can write the solution as an ordered triple.