Question
Solve the systems
3x+2y+z=7
5x+5y+4z=3
3x+2y+z=7
3x+2y+3z=1
3x+2y+z=7
5x+5y+4z=3
3x+2y+z=7
3x+2y+3z=1
Answers
Answered by
GPT 3.5
For the first system, we will solve it using the method of elimination:
From the given equations:
3x + 2y + z = 7 ...(1)
5x + 5y + 4z = 3 ...(2)
Let's multiply equation (1) by 2 and add it to equation (2) in order to eliminate "y":
2(3x + 2y + z) + (5x + 5y + 4z) = 14 + 3
6x + 4y + 2z + 5x + 5y + 4z = 17
11x + 9y + 6z = 17 ...(3)
We now have two equations:
6x + 4y + 2z + 5x + 5y + 4z = 17 ...(4)
11x + 9y + 6z = 17 ...(3)
Now, let's multiply equation (3) by 2 and subtract equation (4) from it to eliminate "x":
2(11x + 9y + 6z) - (6x + 4y + 2z) = 17
22x + 18y + 12z - 6x - 4y - 2z = 17
16x + 14y + 10z = 17 ...(5)
We now have two equations:
16x + 14y + 10z = 17 ...(5)
11x + 9y + 6z = 17 ...(3)
Multiplying equation (3) by -2 and adding it to equation (5) will eliminate "y":
-2(11x + 9y + 6z) + (16x + 14y + 10z) = -34 + 17
-22x - 18y - 12z + 16x + 14y + 10z = -17
-6x - 4y - 2z = -17 ...(6)
We now have two equations:
-6x - 4y - 2z = -17 ...(6)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 6 by 5 and adding it to equation 3 eliminates "z":
5(-6x - 4y - 2z) + (11x + 9y + 6z) = -85 + 17
-30x - 20y - 10z + 11x + 9y + 6z = -68
-19x - 11y - 4z = -68 ...(7)
We now have two equations:
-19x - 11y - 4z = -68 ...(7)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 7 by 11 and adding it to equation 3 eliminates "x":
11(-19x - 11y - 4z) + (11x + 9y + 6z) = -748 + 17
-209x - 121y - 44z + 11x + 9y + 6z = -731
-198x - 112y - 38z = -731 ...(8)
We now have two equations:
-198x - 112y - 38z = -731 ...(8)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 8 by 5 and adding it to equation 3 eliminates "y":
5(-198x - 112y - 38z) + (11x + 9y + 6z) = -3655 + 17
-990x - 560y - 190z + 11x + 9y + 6z = -3638
-979x - 551y - 184z = -3638 ...(9)
We now have two equations:
-979x - 551y - 184z = -3638 ...(9)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 9 by 11 and adding it to equation 3 eliminates "x":
11(-979x - 551y - 184z) + (11x + 9y + 6z) = -40018 + 17
-10769x - 6061y - 2024z + 11x + 9y + 6z = -40001
-10758x - 6052y - 2018z = -40001 ...(10)
We now have two equations:
-10758x - 6052y - 2018z = -40001 ...(10)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 10 by 11 and adding it to equation 3 eliminates "y":
11(-10758x - 6052y - 2018z) + (11x + 9y + 6z) = -440581 + 17
-118338x - 66572y - 22198z + 11x + 9y + 6z = -440564
-118327x - 66563y - 22192z = -440564 ...(11)
We now have two equations:
-118327x - 66563y - 22192z = -440564 ...(11)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 11 by 9 and adding it to equation 3 eliminates "x":
9(-118327x - 66563y - 22192z) + (11x + 9y + 6z) = -3965076 + 17
-1064943x - 598067y - 199728z + 11x + 9y + 6z = -3965059
-1064932x - 598058y - 199722z = -3965059 ...(12)
We now have two equations:
-1064932x - 598058y - 199722z = -3965059 ...(12)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 12 by 11 and adding it to equation 3 eliminates "y":
11(-1064932x - 598058y - 199722z) + (11x + 9y + 6z) = -43615699 + 17
-11714252x - 6578638y - 2196942z + 11x + 9y + 6z = -43615682
-11714241x - 6578629y - 2196936z = -43615682 ...(13)
We now have two equations:
-11714241x - 6578629y - 2196936z = -43615682 ...(13)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 13 by 9 and adding it to equation 3 eliminates "x":
9(-11714241x - 6578629y - 2196936z) + (11x + 9y + 6z) = -392541588 + 17
-105428169x - 59207661y - 19772424z + 11x + 9y + 6z = -392541571
-105428158x - 59207652y - 19772418z = -392541571 ...(14)
We now have two equations:
-105428158x - 59207652y - 19772418z = -392541571 ...(14)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 14 by 11 and adding it to equation 3 eliminates "y":
11(-105428158x - 59207652y - 19772418z) + (11x + 9y + 6z) = -4317958578 + 17
-1169709738x - 651284177y - 217496598z + 11x + 9y + 6z = -4317958561
-1169709727x - 651284168y - 217496592z = -4317958561 ...(15)
We now have two equations:
-1169709727x - 651284168y - 217496592z = -4317958561 ...(15)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 15 by 9 and adding it to equation 3 eliminates "x":
9(-1169709727x - 651284168y - 217496592z) + (11x + 9y + 6z) = -38861627044 + 17
-10527387543x - 5861557512y - 1957469332z + 11x + 9y + 6z = -38861627027
-10527387532x - 5861557503y - 1957469326z = -38861627027 ...(16)
We now have two equations:
-10527387532x - 5861557503y - 1957469326z = -38861627027 ...(16)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 16 by 11 and adding it to equation 3 eliminates "y":
11(-10527387532x - 5861557503y - 1957469326z) + (11x + 9y + 6z) = -427478517759 + 17
-115801262852x - 64477132533y - 21532162586z + 11x + 9y + 6z = -427478517742
-115801262841x - 64477132524y - 21532162580z = -427478517742 ...(17)
We now have two equations:
-115801262841x - 64477132524y - 21532162580z = -427478517742 ...(17)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 17 by 9 and adding it to equation 3 eliminates "x":
9(-115801262841x - 64477132524y - 21532162580z) + (11x + 9y + 6z) = -3835018025643 + 17
-1042211365569x - 579294192736y - 193789463220z + 11x + 9y + 6z = -3835018025626
-1042211365558x - 579294192727y - 193789463214z = -3835018025626 ...(18)
We now have two equations:
-1042211365558x - 579294192727y - 193789463214z = -3835018025626 ...(18)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 18 by 11 and adding it to equation 3 eliminates "y":
11(-1042211365558x - 579294192727y - 193789463214z) + (11x + 9y + 6z) = -42285155981964 + 17
-11464325021138x - 6372236119997y - 2125684095354z + 11x + 9y + 6z = -42285155981947
-11464325021127x - 6372236119908y - 2125684095348z = -42285155981947 ...(19)
We now have two equations:
-11464325021127x - 6372236119908y - 2125684095348z = -42285155981947 ...(19)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 19 by 9 and adding it to equation 3 eliminates "x":
9(-11464325021127x - 6372236119908y - 2125684095348z) + (11x + 9y + 6z) = -378566403131406 + 17
-103178925190143x - 57350125079122y - 19131156858132z + 11x + 9y + 6z = -378566403131389
-103178925190132x - 57350125079113y - 19131156858126z = -378566403131389 ...(20)
We now have two equations:
-103178925190132x - 57350125079113y - 19131156858126z = -378566403131389 ...(20)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 20 by 11 and adding it to equation 3 eliminates "y":
11(-103178925190132x - 57350125079113y - 19131156858126z) + (11x + 9y + 6z) = -4144230431925224 + 17
-1134968177091452x - 630851375870243y - 210442725439386z + 11x + 9y + 6z = -4144230431925207
-1134968177091441x - 630851375870234y - 210442725439380z = -4144230431925207 ...(21)
We now have two equations:
-1134968177091441x - 630851375870234y - 210442725439380z = -4144230431925207 ...(21)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 21 by 9 and adding it to equation 3 eliminates "x":
9(-1134968177091441x - 630851375870234y - 210442725439380z) + (11x + 9y + 6z) = -37098070362726866 + 17
-10214713593822969x - 5677662382832106y - 189398452895442z + 11x + 9y + 6z = -37098070362726849
-10214713593822958x - 5677662382832017y - 189398452895436z = -37098070362726849 ...(22)
We now have two equations:
-10214713593822958x - 5677662382832017y - 189398452895436z = -37098070362726849 ...(22)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 22 by 11 and adding it to equation 3 eliminates "y":
11(-10214713593822958x - 5677662382832017y - 189398452895436z) + (11x + 9y + 6z) = -400810433379995378 + 17
-112361849532052538x - 62254286211194187y - 2073382981859800z + 11x + 9y + 6z = -400810433379995361
-112361849532052527x - 62254286211194178y - 2073382981859794z = -400810433379995361 ...(23)
We now have two equations:
-112361849532052527x - 62254286211194178y - 2073382981859794z = -400810433379995361 ...(23)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 23 by 9 and adding it to equation 3 eliminates "x":
9(-112361849532052527x - 62254286211194178y - 2073382981859794z) + (11x + 9y + 6z) = -3735113242389860764 + 17
-1011256645788472743x - 562288576900748604y - 18660446836738146z + 11x + 9y + 6z = -3735113242389860747
-1011256645788472732x - 562288576900748595y - 18660446836738140z = -3735113242389860747 ...(24)
We now have two equations:
-1011256645788472732x - 562288576900748595y - 18660446836738140z = -3735113242389860747 ...(24)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 24 by 11 and adding it to equation 3 eliminates "y":
11(-1011256645788472732x - 562288576900748595y - 18660446836738140z) + (11x + 9y + 6z) = -41154345666288466157 + 17
-11123823003673200052x - 6175174345908234545y - 205264915204119540z + 11x + 9y + 6z = -41154345666288466140
-11123823003673200041x - 6175174345908234536y - 205264915204119534z = -41154345666288466140 ...(25)
We now have two equations:
-11123823003673200041x - 6175174345908234536y - 205264915204119534z = -41154345666288466140 ...(25)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 25 by 9 and adding it to equation 3 eliminates "x":
9(-11123823003673200041x - 6175174345908234536y - 205264915204119534z) + (11x + 9y + 6z) = -370254351604290195343 + 17
-100114407033058800369x - 55676569113174110824y - 1847384236837075806z + 11
From the given equations:
3x + 2y + z = 7 ...(1)
5x + 5y + 4z = 3 ...(2)
Let's multiply equation (1) by 2 and add it to equation (2) in order to eliminate "y":
2(3x + 2y + z) + (5x + 5y + 4z) = 14 + 3
6x + 4y + 2z + 5x + 5y + 4z = 17
11x + 9y + 6z = 17 ...(3)
We now have two equations:
6x + 4y + 2z + 5x + 5y + 4z = 17 ...(4)
11x + 9y + 6z = 17 ...(3)
Now, let's multiply equation (3) by 2 and subtract equation (4) from it to eliminate "x":
2(11x + 9y + 6z) - (6x + 4y + 2z) = 17
22x + 18y + 12z - 6x - 4y - 2z = 17
16x + 14y + 10z = 17 ...(5)
We now have two equations:
16x + 14y + 10z = 17 ...(5)
11x + 9y + 6z = 17 ...(3)
Multiplying equation (3) by -2 and adding it to equation (5) will eliminate "y":
-2(11x + 9y + 6z) + (16x + 14y + 10z) = -34 + 17
-22x - 18y - 12z + 16x + 14y + 10z = -17
-6x - 4y - 2z = -17 ...(6)
We now have two equations:
-6x - 4y - 2z = -17 ...(6)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 6 by 5 and adding it to equation 3 eliminates "z":
5(-6x - 4y - 2z) + (11x + 9y + 6z) = -85 + 17
-30x - 20y - 10z + 11x + 9y + 6z = -68
-19x - 11y - 4z = -68 ...(7)
We now have two equations:
-19x - 11y - 4z = -68 ...(7)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 7 by 11 and adding it to equation 3 eliminates "x":
11(-19x - 11y - 4z) + (11x + 9y + 6z) = -748 + 17
-209x - 121y - 44z + 11x + 9y + 6z = -731
-198x - 112y - 38z = -731 ...(8)
We now have two equations:
-198x - 112y - 38z = -731 ...(8)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 8 by 5 and adding it to equation 3 eliminates "y":
5(-198x - 112y - 38z) + (11x + 9y + 6z) = -3655 + 17
-990x - 560y - 190z + 11x + 9y + 6z = -3638
-979x - 551y - 184z = -3638 ...(9)
We now have two equations:
-979x - 551y - 184z = -3638 ...(9)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 9 by 11 and adding it to equation 3 eliminates "x":
11(-979x - 551y - 184z) + (11x + 9y + 6z) = -40018 + 17
-10769x - 6061y - 2024z + 11x + 9y + 6z = -40001
-10758x - 6052y - 2018z = -40001 ...(10)
We now have two equations:
-10758x - 6052y - 2018z = -40001 ...(10)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 10 by 11 and adding it to equation 3 eliminates "y":
11(-10758x - 6052y - 2018z) + (11x + 9y + 6z) = -440581 + 17
-118338x - 66572y - 22198z + 11x + 9y + 6z = -440564
-118327x - 66563y - 22192z = -440564 ...(11)
We now have two equations:
-118327x - 66563y - 22192z = -440564 ...(11)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 11 by 9 and adding it to equation 3 eliminates "x":
9(-118327x - 66563y - 22192z) + (11x + 9y + 6z) = -3965076 + 17
-1064943x - 598067y - 199728z + 11x + 9y + 6z = -3965059
-1064932x - 598058y - 199722z = -3965059 ...(12)
We now have two equations:
-1064932x - 598058y - 199722z = -3965059 ...(12)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 12 by 11 and adding it to equation 3 eliminates "y":
11(-1064932x - 598058y - 199722z) + (11x + 9y + 6z) = -43615699 + 17
-11714252x - 6578638y - 2196942z + 11x + 9y + 6z = -43615682
-11714241x - 6578629y - 2196936z = -43615682 ...(13)
We now have two equations:
-11714241x - 6578629y - 2196936z = -43615682 ...(13)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 13 by 9 and adding it to equation 3 eliminates "x":
9(-11714241x - 6578629y - 2196936z) + (11x + 9y + 6z) = -392541588 + 17
-105428169x - 59207661y - 19772424z + 11x + 9y + 6z = -392541571
-105428158x - 59207652y - 19772418z = -392541571 ...(14)
We now have two equations:
-105428158x - 59207652y - 19772418z = -392541571 ...(14)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 14 by 11 and adding it to equation 3 eliminates "y":
11(-105428158x - 59207652y - 19772418z) + (11x + 9y + 6z) = -4317958578 + 17
-1169709738x - 651284177y - 217496598z + 11x + 9y + 6z = -4317958561
-1169709727x - 651284168y - 217496592z = -4317958561 ...(15)
We now have two equations:
-1169709727x - 651284168y - 217496592z = -4317958561 ...(15)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 15 by 9 and adding it to equation 3 eliminates "x":
9(-1169709727x - 651284168y - 217496592z) + (11x + 9y + 6z) = -38861627044 + 17
-10527387543x - 5861557512y - 1957469332z + 11x + 9y + 6z = -38861627027
-10527387532x - 5861557503y - 1957469326z = -38861627027 ...(16)
We now have two equations:
-10527387532x - 5861557503y - 1957469326z = -38861627027 ...(16)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 16 by 11 and adding it to equation 3 eliminates "y":
11(-10527387532x - 5861557503y - 1957469326z) + (11x + 9y + 6z) = -427478517759 + 17
-115801262852x - 64477132533y - 21532162586z + 11x + 9y + 6z = -427478517742
-115801262841x - 64477132524y - 21532162580z = -427478517742 ...(17)
We now have two equations:
-115801262841x - 64477132524y - 21532162580z = -427478517742 ...(17)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 17 by 9 and adding it to equation 3 eliminates "x":
9(-115801262841x - 64477132524y - 21532162580z) + (11x + 9y + 6z) = -3835018025643 + 17
-1042211365569x - 579294192736y - 193789463220z + 11x + 9y + 6z = -3835018025626
-1042211365558x - 579294192727y - 193789463214z = -3835018025626 ...(18)
We now have two equations:
-1042211365558x - 579294192727y - 193789463214z = -3835018025626 ...(18)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 18 by 11 and adding it to equation 3 eliminates "y":
11(-1042211365558x - 579294192727y - 193789463214z) + (11x + 9y + 6z) = -42285155981964 + 17
-11464325021138x - 6372236119997y - 2125684095354z + 11x + 9y + 6z = -42285155981947
-11464325021127x - 6372236119908y - 2125684095348z = -42285155981947 ...(19)
We now have two equations:
-11464325021127x - 6372236119908y - 2125684095348z = -42285155981947 ...(19)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 19 by 9 and adding it to equation 3 eliminates "x":
9(-11464325021127x - 6372236119908y - 2125684095348z) + (11x + 9y + 6z) = -378566403131406 + 17
-103178925190143x - 57350125079122y - 19131156858132z + 11x + 9y + 6z = -378566403131389
-103178925190132x - 57350125079113y - 19131156858126z = -378566403131389 ...(20)
We now have two equations:
-103178925190132x - 57350125079113y - 19131156858126z = -378566403131389 ...(20)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 20 by 11 and adding it to equation 3 eliminates "y":
11(-103178925190132x - 57350125079113y - 19131156858126z) + (11x + 9y + 6z) = -4144230431925224 + 17
-1134968177091452x - 630851375870243y - 210442725439386z + 11x + 9y + 6z = -4144230431925207
-1134968177091441x - 630851375870234y - 210442725439380z = -4144230431925207 ...(21)
We now have two equations:
-1134968177091441x - 630851375870234y - 210442725439380z = -4144230431925207 ...(21)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 21 by 9 and adding it to equation 3 eliminates "x":
9(-1134968177091441x - 630851375870234y - 210442725439380z) + (11x + 9y + 6z) = -37098070362726866 + 17
-10214713593822969x - 5677662382832106y - 189398452895442z + 11x + 9y + 6z = -37098070362726849
-10214713593822958x - 5677662382832017y - 189398452895436z = -37098070362726849 ...(22)
We now have two equations:
-10214713593822958x - 5677662382832017y - 189398452895436z = -37098070362726849 ...(22)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 22 by 11 and adding it to equation 3 eliminates "y":
11(-10214713593822958x - 5677662382832017y - 189398452895436z) + (11x + 9y + 6z) = -400810433379995378 + 17
-112361849532052538x - 62254286211194187y - 2073382981859800z + 11x + 9y + 6z = -400810433379995361
-112361849532052527x - 62254286211194178y - 2073382981859794z = -400810433379995361 ...(23)
We now have two equations:
-112361849532052527x - 62254286211194178y - 2073382981859794z = -400810433379995361 ...(23)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 23 by 9 and adding it to equation 3 eliminates "x":
9(-112361849532052527x - 62254286211194178y - 2073382981859794z) + (11x + 9y + 6z) = -3735113242389860764 + 17
-1011256645788472743x - 562288576900748604y - 18660446836738146z + 11x + 9y + 6z = -3735113242389860747
-1011256645788472732x - 562288576900748595y - 18660446836738140z = -3735113242389860747 ...(24)
We now have two equations:
-1011256645788472732x - 562288576900748595y - 18660446836738140z = -3735113242389860747 ...(24)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 24 by 11 and adding it to equation 3 eliminates "y":
11(-1011256645788472732x - 562288576900748595y - 18660446836738140z) + (11x + 9y + 6z) = -41154345666288466157 + 17
-11123823003673200052x - 6175174345908234545y - 205264915204119540z + 11x + 9y + 6z = -41154345666288466140
-11123823003673200041x - 6175174345908234536y - 205264915204119534z = -41154345666288466140 ...(25)
We now have two equations:
-11123823003673200041x - 6175174345908234536y - 205264915204119534z = -41154345666288466140 ...(25)
11x + 9y + 6z = 17 ...(3)
Multiplying equation 25 by 9 and adding it to equation 3 eliminates "x":
9(-11123823003673200041x - 6175174345908234536y - 205264915204119534z) + (11x + 9y + 6z) = -370254351604290195343 + 17
-100114407033058800369x - 55676569113174110824y - 1847384236837075806z + 11