Finding solutions of a system of linear equations 2 x 1 − 4 x 2 + 8 x 4 = 10 3 x 1 + 2 x 2 = 3 3 x 1 + 2 x 3 + 6 x 4 = 11 8 x 1 + 6 x 2 + 2 x 3 + 14 x 4 = 24 Find the solution set W. Is W a vector space?I know I could express one variable from one of the equations and substitute into the remaining ones. For example, from the second equation I get x 1 = 1 − 2 3 x 2 .After substituting this into the remaining equations I have − 16 3 x 2 + 8 x 4 = 8 − 2 x 2 + 2 x 3 + 6 x 4 = 7 2 3 x 2 + 2 x 3 + 14 x 4 = 16 I could continue by expressing another variable and plug into other equations. But this approach seems to be rather cumbersome and if I have more than three equations I am prone to make mistakes. Is there a more elegant way to do this?

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Finding solutions of a system of linear equations                    2                  x          1                −        4                  x          2                +        8                  x          4                                    =        10                            3                  x          1                +        2                  x          2                                    =        3                            3                  x          1                +        2                  x          3                +        6                  x          4                                    =        11                            8                  x          1                +        6                  x          2                +        2                  x          3                +        14                  x          4                                    =        24            Find the solution set   W. Is   W a vector space?I know I could express one variable from one of the equations and substitute into the remaining ones. For example, from the second equation I get       x    1    =  1  −      2    3        x    2  .After substituting this into the remaining equations I have                    −                  16          3                          x          2                +        8                  x          4                                    =        8                            −        2                  x          2                +        2                  x          3                +        6                  x          4                                    =        7                                      2          3                          x          2                +        2                  x          3                +        14                  x          4                                    =        16            I could continue by expressing another variable and plug into other equations. But this approach seems to be rather cumbersome and if I have more than three equations I am prone to make mistakes. Is there a more elegant way to do this?

trajeronls · Accepted Answer

If you set up the augmented matrix for the system and perform RREF (Gaussian Elimination), you end up with:      [                            1                          0                          0                          −          2                          −          1                                      0                          1                          0                          3                          3                                      0                          0                          1                          6                          7                                      0                          0                          0                          0                          0                      ]  There are other methods (substitution, inverses, ....), but you did not indicate what you are currently familiar with.For example, if using substitution, here are hints:Use the second equation to solve for       x    1   and substitute that in to the first.Solve the first equation for       x    4   and substitute into the third equation.Solve the third equation for       x    3   and substitute into the fourth equation.Solve the fourth equation for       x    2   and then solve the system.You should get the same answers I show in my augmented matrix.I&#039;ll let you figure out the details and answer for the solution   W and is   W a vector space.

gnatopoditw · Accepted Answer

Good answer

Finding solutions of a system of linear equations <mtable columnalign="right left right left rig

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Answer & Explanation

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