Elimination: 10441

Solve the linear system of equations by Gaussian elimination:

3x1 −2x2 −5x3 + x4 = 3
2x1 −3x2 + x3 + 5x4 = −3
x1 + 2x2 −4x4 = −3
x1 - x2 −4x3 + 9x4 = 22

Correct answer:

x1 =  -1
x2 =  3
x3 =  -2
x4 =  2

Step-by-step explanation:


3x1 −2x2 −5x3 + x4 = 3
2x1 −3x2 + x3 +5x4 = −3
x1 +2x2 −4x4 = −3
x1 − x2 −4x3 +9x4 = 22

3·x1 -2·x2 -5·x3 + x4 = 3
2·x1 -3·x2 + x3 +5·x4 = -3
x1 +2·x2 -4·x4 = -3
x1 - x2 -4·x3 +9·x4 = 22

3x1-2x2-5x3+x4 = 3
2x1-3x2+x3+5x4 = -3
x1+2x2-4x4 = -3
x1-x2-4x3+9x4 = 22

Row 2 - 2/3 · Row 1 → Row 2
3x1-2x2-5x3+x4 = 3
-1.6667x2+4.3333x3+4.3333x4 = -5
x1+2x2-4x4 = -3
x1-x2-4x3+9x4 = 22

Row 3 - 1/3 · Row 1 → Row 3
3x1-2x2-5x3+x4 = 3
-1.6667x2+4.3333x3+4.3333x4 = -5
2.6667x2+1.6667x3-4.3333x4 = -4
x1-x2-4x3+9x4 = 22

Row 4 - 1/3 · Row 1 → Row 4
3x1-2x2-5x3+x4 = 3
-1.6667x2+4.3333x3+4.3333x4 = -5
2.6667x2+1.6667x3-4.3333x4 = -4
-0.3333x2-2.3333x3+8.6667x4 = 21

Pivot: Row 2 ↔ Row 3
3x1-2x2-5x3+x4 = 3
2.6667x2+1.6667x3-4.3333x4 = -4
-1.6667x2+4.3333x3+4.3333x4 = -5
-0.3333x2-2.3333x3+8.6667x4 = 21

Row 3 - -1.66666667/2.66666667 · Row 2 → Row 3
3x1-2x2-5x3+x4 = 3
2.6667x2+1.6667x3-4.3333x4 = -4
5.375x3+1.625x4 = -7.5
-0.3333x2-2.3333x3+8.6667x4 = 21

Row 4 - -0.33333333/2.66666667 · Row 2 → Row 4
3x1-2x2-5x3+x4 = 3
2.6667x2+1.6667x3-4.3333x4 = -4
5.375x3+1.625x4 = -7.5
-2.125x3+8.125x4 = 20.5

Row 4 - -2.125/5.375 · Row 3 → Row 4
3x1-2x2-5x3+x4 = 3
2.6667x2+1.6667x3-4.3333x4 = -4
5.375x3+1.625x4 = -7.5
8.7674x4 = 17.5349


x4 = 17.53488372/8.76744186 = 2
x3 = -7.5-1.625x4/5.375 = -7.5-1.625 · 2/5.375 = -2
x2 = -4-1.6666666666667x3+4.3333333333333x4/2.66666667 = -4-1.66666667 · (-2)+4.33333333 · 2/2.66666667 = 3
x1 = 3+2x2+5x3-x4/3 = 3+2 · 3+5 · (-2)-2/3 = -1

x1 = -1
x2 = 3
x3 = -2
x4 = 2

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