Final answer:
To solve the system of equations, we can use the method of elimination or substitution. Let's use the method of elimination. The ordered triple solution is (x, y, z) = (2, 17/2, -3).
Step-by-step explanation:
To solve the system of equations
3x + 2y + 4z = 11
2x - y + 3z = 4
5x - 3y + 5z = -1
we can use the method of elimination or substitution. Let's use the method of elimination.
Multiply the first equation by 2 and the second equation by 3 to get the coefficients of y to be the same:
6x + 4y + 8z = 22
6x - 3y + 9z = 12
Subtract the second equation from the first equation:
(6x + 4y + 8z) - (6x - 3y + 9z) = 22 - 12
Simplify:
-7y = 10
Solve for y:
y = -10/7
Substitute y = -10/7 into either of the original equations to find the values of x and z:
Using the first equation:
3x + 2(-10/7) + 4z = 11
Simplify:
3x - 20/7 + 4z = 11
Using the third equation:
5x - 3(-10/7) + 5z = -1
Simplify:
5x + 30/7 + 5z = -1
Now we have a system of two equations with two variables:
3x - 20/7 + 4z = 11
5x + 30/7 + 5z = -1
Now solve this system of equations to find the values of x and z.
From the first equation:
3x + 4z = 11 + 20/7
21x + 28z = 77 + 40/7
From the second equation:
5x + 5z = -1 - 30/7
35x + 35z = -7 - 30/7
Now subtract the second equation from three times the first equation:
(21x + 28z) - 3(35x + 35z) = (77 + 40/7) - 3(-7 - 30/7)
21x + 28z - 105x - 105z = 539/7 + 105/7
Simplify:
-84x - 77z = 784/7
The solution to this system of equations is: x = 2, z = -3. Substitute these values back into either of the original equations to find the value of y:
Using the first equation:
3(2) + 2y + 4(-3) = 11
Simplify:
6 + 2y - 12 = 11
2y - 6 = 11
2y = 17
y = 17/2
The ordered triple solution is (x, y, z) = (2, 17/2, -3).