document.write( "Question 1205169: Consider the following system of linear equations:
\n" ); document.write( "𝑥 + 3𝑦 + 2𝑧 − 𝑤 = −1
\n" ); document.write( "−3𝑥 − 7𝑦 + (𝑝 − 6)𝑧 + 2𝑤 = 1
\n" ); document.write( "2𝑥 + 𝑝^2𝑧 + 𝑝𝑤 = 𝑞^2\r
\n" ); document.write( "\n" ); document.write( "where 𝑝 and 𝑞 are real numbers.
\n" ); document.write( "Using Gaussian elimination, determine all possible values of 𝑝 and 𝑞 such that the system has infinitely many solutions with two free variables and solve the system. \n" ); document.write( "

Algebra.Com's Answer #845442 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "The augmented matrix for that system of equations is:\r\n" );
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document.write( "Now if you had to find the row-reduced echelon form of that matrix by hand,\r\n" );
document.write( "that would be a big chore. But these days, we are getting more and more\r\n" );
document.write( "specialized calculators in sites online.  So we are having to do less and less\r\n" );
document.write( "mathematical manipulations ourselves.  However, you can bet your sweet bippy\r\n" );
document.write( "that sooner or later you're going to have to start paying out good money to\r\n" );
document.write( "subscribe to these sites.  But while they're still free, get the row-reduced\r\n" );
document.write( "echelon form of this matrix from this site, which hasn't started charging yet:\r\n" );
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document.write( "https://www.emathhelp.net/en/calculators/linear-algebra/reduced-row-echelon-form-rref-calculator/\r\n" );
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document.write( "This shows that:\r\n" );
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document.write( "So the system will always have a solution, and infinitely many solutions, as\r\n" );
document.write( "long as none of those denominators equal 0. That is,\r\n" );
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document.write( "There are 2 free variables, w and q, and there's almost a 3rd free variable,\r\n" );
document.write( "because p is free to take on any value other than 1 and -4.  So p cannot be\r\n" );
document.write( "considered totally free.\r\n" );
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document.write( "Edwin

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