Solve the system of equations using matrices. Use Gaussian elimination with back- substitution.
x+y+z = -5
x-y+3z = -1
4x+y+z = -2
To solve the given system of equations using matrices and the Gaussian elimination method with back-substitution, we'll represent the system of equations in augmented matrix form.
Step 1: Write the augmented matrix:
[1 1 1 | -5]
[1 -1 3 | -1]
[4 1 1 | -2]
Step 2: Perform row operations to eliminate the x-coefficient in the second and third equations:
R2 = R2 - R1
R3 = R3 - 4R1
New matrix:
[1 1 1 | -5]
[0 -2 2 | 4]
[0 -3 -3 | 18]
Step 3: Perform row operations to eliminate the y-coefficient in the third equation:
R3 = R3 - (3/2)R2
New matrix:
[1 1 1 | -5]
[0 -2 2 | 4]
[0 0 -3 | 9]
Step 4: Solve for z by applying back-substitution:
-3z = 9 -> z = -3
Step 5: Substituting the value of z back into the second equation, solve for y:
-2y + 2z = 4
-2y + 2(-3) = 4
-2y - 6 = 4
-2y = 10
y = -5
Step 6: Finally, substitute the values of y and z back into the first equation to solve for x:
x + y + z = -5
x + (-5) + (-3) = -5
x - 8 = -5
x = 3
Therefore, the solution to the given system of equations is x = 3, y = -5, and z = -3.