The question requires to use gaussian elimination to solve x,y,z:
2x-6y-z=7
x+2y+z=0
x+4y+2z=-3
The answer i got is x=-6, y=-4, z=5
But the answer from answersheet is x=3, y=1/2, z=-4
Is it possible to get a different answer using gaussian elimination? Or there is only an exact answer.
i found my mistake and got the answer. Thanks!
Ok, I ran it through
Wolfram and also got the published answer
www.wolframalpha.com/input?i=2x-6y-z%3D7%2C+x%2B2y%2Bz%3D0%2C+x%2B4y%2B2z%3D-3
There is only one solution to the system, regardless of the method used.
did you check your answer to see whether it works in the equations?
2x-6y-z=7
2(-6)-6(-4)-5 = 7 ✅
x+2y+z=0
(-6)+2(-4)+5 = -9 ❌
x+4y+2z=-3
(-6)+4(-4)+2(5) = -12 ❌
ALWAYS check your answer!
While Gaussian elimination is a systematic method for solving systems of linear equations, it is possible to make errors or encounter inconsistencies that could lead to different final answers. It's important to double-check your work to ensure accuracy.
To perform Gaussian elimination, we'll follow these steps:
Step 1: Write the augmented matrix of the system:
[ 2 -6 -1 | 7 ]
[ 1 2 1 | 0 ]
[ 1 4 2 | -3 ]
Step 2: Apply row operations to convert the matrix into upper triangular form:
R2 = R2 - (1/2)R1
R3 = R3 - (1/2)R1
The updated matrix:
[ 2 -6 -1 | 7 ]
[ 0 5 3 | -3.5 ]
[ 0 7 3 | -6.5 ]
Step 3: Further simplify the matrix:
R3 = R3 - (7/5)R2
The updated matrix:
[ 2 -6 -1 | 7 ]
[ 0 5 3 | -3.5 ]
[ 0 0 -4 | -1.2 ]
Step 4: Solve for z:
-4z = -1.2
z = (-1.2) / (-4) = 0.3
Step 5: Substitute z = 0.3 into the second equation and solve for y:
5y + 3(0.3) = -3.5
5y + 0.9 = -3.5
5y = -4.4
y = (-4.4) / 5 = -0.88 or -22/25
Step 6: Substitute y = -0.88 into the first equation and solve for x:
2x - 6(-0.88) - 0.3 = 7
2x + 5.28 - 0.3 = 7
2x + 4.98 = 7
2x = 2.02
x = 2.02 / 2 = 1.01
Therefore, the solution to the system of equations is x ≈ 1.01, y ≈ -0.88, z ≈ 0.3.
Comparing this to the answer sheet, it seems your initial answer of x = -6, y = -4, z = 5 is incorrect.