document.write( "Question 1122649: Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
\n" ); document.write( "2x1 − x2 + x4 = 9
\n" ); document.write( "3x2 − x3 − x4 = −9
\n" ); document.write( "x1 + x3 − 3x4 = −2
\n" ); document.write( "x1 + x2 + 2x3 = 0
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Algebra.Com's Answer #738812 by ikleyn(52781) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "The matrix form is\r\n" );
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document.write( " .  = .\r\n" );
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document.write( "The rule is simple: where the unknown is missed (omitted) in the current equation, you put zero coefficient to the matrix.\r\n" );
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document.write( "To solve it, I used free of charge online solver\r\n" );
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document.write( "https://matrix.reshish.com/gaussSolution.php\r\n" );
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document.write( "The solution process and the results are shown below:\r\n" );
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document.write( "Your matrix\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	2	-1	0	1	9\r\n" );
document.write( "2	0	3	-1	-1	-9\r\n" );
document.write( "3	1	0	1	-3	-2\r\n" );
document.write( "4	1	1	2	0	0\r\n" );
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document.write( "Find the pivot in the 1st column and swap the 3rd and the 1st rows\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	1	-3	-2\r\n" );
document.write( "2	0	3	-1	-1	-9\r\n" );
document.write( "3	2	-1	0	1	9\r\n" );
document.write( "4	1	1	2	0	0\r\n" );
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document.write( "Multiply the 1st row by 2\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	2	0	2	-6	-4\r\n" );
document.write( "2	0	3	-1	-1	-9\r\n" );
document.write( "3	2	-1	0	1	9\r\n" );
document.write( "4	1	1	2	0	0\r\n" );
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document.write( "Subtract the 1st row from the 3rd row and restore it\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	1	-3	-2\r\n" );
document.write( "2	0	3	-1	-1	-9\r\n" );
document.write( "3	0	-1	-2	7	13\r\n" );
document.write( "4	1	1	2	0	0\r\n" );
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document.write( "Subtract the 1st row from the 4th row\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	1	-3	-2\r\n" );
document.write( "2	0	3	-1	-1	-9\r\n" );
document.write( "3	0	-1	-2	7	13\r\n" );
document.write( "4	0	1	1	3	2\r\n" );
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document.write( "Find the pivot in the 2nd column (inversing the sign in the whole row) and swap the 3rd and the 2nd rows\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	1	-3	-2\r\n" );
document.write( "2	0	1	2	-7	-13\r\n" );
document.write( "3	0	3	-1	-1	-9\r\n" );
document.write( "4	0	1	1	3	2\r\n" );
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document.write( "Multiply the 2nd row by 3\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	1	-3	-2\r\n" );
document.write( "2	0	3	6	-21	-39\r\n" );
document.write( "3	0	3	-1	-1	-9\r\n" );
document.write( "4	0	1	1	3	2\r\n" );
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document.write( "Subtract the 2nd row from the 3rd row and restore it\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	1	-3	-2\r\n" );
document.write( "2	0	1	2	-7	-13\r\n" );
document.write( "3	0	0	-7	20	30\r\n" );
document.write( "4	0	1	1	3	2\r\n" );
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document.write( "Subtract the 2nd row from the 4th row\r\n" );
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document.write( "      X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	1	-3	-2\r\n" );
document.write( "2	0	1	2	-7	-13\r\n" );
document.write( "3	0	0	-7	20	30\r\n" );
document.write( "4	0	0	-1	10	15\r\n" );
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document.write( "Find the pivot in the 3rd column (inversing the sign in the whole row) and swap the 4th and the 3rd rows\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	1	-3	-2\r\n" );
document.write( "2	0	1	2	-7	-13\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	-7	20	30\r\n" );
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document.write( "Subtract the 3rd row from the 1st row\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	7	13\r\n" );
document.write( "2	0	1	2	-7	-13\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	-7	20	30\r\n" );
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document.write( "Multiply the 3rd row by 2\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	7	13\r\n" );
document.write( "2	0	1	2	-7	-13\r\n" );
document.write( "3	0	0	2	-20	-30\r\n" );
document.write( "4	0	0	-7	20	30\r\n" );
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document.write( "Subtract the 3rd row from the 2nd row and restore it\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	7	13\r\n" );
document.write( "2	0	1	0	13	17\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	-7	20	30\r\n" );
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document.write( "Multiply the 3rd row by -7\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	7	13\r\n" );
document.write( "2	0	1	0	13	17\r\n" );
document.write( "3	0	0	-7	70	105\r\n" );
document.write( "4	0	0	-7	20	30\r\n" );
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document.write( "Subtract the 3rd row from the 4th row and restore it\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	7	13\r\n" );
document.write( "2	0	1	0	13	17\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	0	-50	-75\r\n" );
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document.write( "Make the pivot in the 4th column by dividing the 4th row by -50\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	7	13\r\n" );
document.write( "2	0	1	0	13	17\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	0	1	3/2\r\n" );
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document.write( "Multiply the 4th row by 7\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	7	13\r\n" );
document.write( "2	0	1	0	13	17\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	0	7	21/2\r\n" );
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document.write( "Subtract the 4th row from the 1st row and restore it\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	0	5/2\r\n" );
document.write( "2	0	1	0	13	17\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	0	1	3/2\r\n" );
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document.write( "Multiply the 4th row by 13\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	0	5/2\r\n" );
document.write( "2	0	1	0	13	17\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	0	13	39/2\r\n" );
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document.write( "Subtract the 4th row from the 2nd row and restore it\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	0	5/2\r\n" );
document.write( "2	0	1	0	0	-5/2\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	0	1	3/2\r\n" );
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document.write( "Multiply the 4th row by -10\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	0	5/2\r\n" );
document.write( "2	0	1	0	0	-5/2\r\n" );
document.write( "3	0	0	1	-10	-15\r\n" );
document.write( "4	0	0	0	-10	-15\r\n" );
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document.write( "Subtract the 4th row from the 3rd row and restore it\r\n" );
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document.write( "       X1	X2	X3	X4	b\r\n" );
document.write( "1	1	0	0	0	5/2\r\n" );
document.write( "2	0	1	0	0	-5/2\r\n" );
document.write( "3	0	0	1	0	0\r\n" );
document.write( "4	0	0	0	1	3/2\r\n" );
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document.write( "Solution set:\r\n" );
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document.write( "x1 = 5/2\r\n" );
document.write( "x2 = -5/2\r\n" );
document.write( "x3 = 0\r\n" );
document.write( "x4 = 3/2\r\n" );
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