document.write( "Question 1069422: Need help solving system with three variables\r
\n" ); document.write( "\n" ); document.write( "-2x + 8y + 2z = 4
\n" ); document.write( " x + 6y + 3z = 4
\n" ); document.write( " 3x - 2y + z = 0 \n" ); document.write( "

Algebra.Com's Answer #684622 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Need help solving system $\"highlight%28cross%28with%29%29\"$ in three variables\r
\n" ); document.write( "\n" ); document.write( "-2x + 8y + 2z = 4
\n" ); document.write( " x + 6y + 3z = 4
\n" ); document.write( " 3x - 2y + z = 0
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document.write( "-2x + 8y + 2z = 4    (1)\r\n" );
document.write( "  x + 6y + 3z = 4    (2)\r\n" );
document.write( " 3x - 2y +  z = 0    (3)\r\n" );
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document.write( "Add equations (1) and (3) (both sides). You will get\r\n" );
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document.write( "  x + 6y + 3z = 4    (4).\r\n" );
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document.write( "Now compare equation (4) with equation (2). What do you see ??\r\n" );
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document.write( "Their left sides are identical. Their right sides are identical, too.\r\n" );
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document.write( "What does it mean ??  - It means that the equation (2) is DEPENDENT on equations (1) and (3).\r\n" );
document.write( "In other words, equation (2) doesn't carry new information.\r\n" );
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document.write( "What does it mean ??  - It means that, factually, there are only TWO equations for THREE unknowns.\r\n" );
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document.write( "What does it mean ??  - It means that the original system of equations HAS INFINITELY MANY solutions.\r\n" );
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document.write( "Answer. The system of equations (1), (2), (3) HAS INFINITELY MANY solutions.\r\n" );
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\n" ); document.write( "\n" ); document.write( "There is a bunch of lessons on solving systems of linear equations in three unknowns by the Substitution and Elimination methods \r
\n" ); document.write( "\n" ); document.write( "    - Solving systems of linear equations in 3 unknowns by the Substitution method \r
\n" ); document.write( "\n" ); document.write( "    - BRIEFLY on solving systems of linear equations in 3 unknowns by the Substitution method \r
\n" ); document.write( "\n" ); document.write( "    - Solving systems of linear equations in 3 unknowns by the Elimination method \r
\n" ); document.write( "\n" ); document.write( "    - BRIEFLY on solving systems of linear equations in 3 unknowns by the Elimination method\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of LESSONS on solving systems of linear equations in three unknowns by the Substitution and/or Elimination methods\r
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\n" ); document.write( "\n" ); document.write( "On Cramer's rule for solving systems of 3 equations in 3 unknowns see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Determinant of a 3x3 matrix \r
\n" ); document.write( "\n" ); document.write( "    - Co-factoring the determinant of a 3x3 matrix \r
\n" ); document.write( "\n" ); document.write( "    - HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)\r
\n" ); document.write( "\n" ); document.write( "    - Solving systems of linear equations in three unknowns using determinant (Cramer's rule)\r
\n" ); document.write( "\n" ); document.write( "    - Solving word problems by reducing to systems of linear equations in three unknowns\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic
\n" ); document.write( "     \"3x3-Matrices, determinants, Cramer's rule for systems in three unknowns\" \r
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