document.write( "Question 798071: Consider \r
\n" ); document.write( "\n" ); document.write( "4x+tx−20y = 0 \r
\n" ); document.write( "\n" ); document.write( "x−4y+ty = 0\r
\n" ); document.write( "\n" ); document.write( "Find an expression for the determinant of the coefficient matrix A of this system. \r
\n" ); document.write( "\n" ); document.write( "det(A) = ? \r
\n" ); document.write( "\n" ); document.write( "Use your previous answer to give the solutions for x and y of this system for all real values of t. \r
\n" ); document.write( "\n" ); document.write( "x= ? and x= ?\r
\n" ); document.write( "\n" ); document.write( "*** My problem is that I don't understand how to begin it. We usually work with 3*3 or 4*4 matrices in class. \r
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\n" ); document.write( "Cristina \n" ); document.write( "

Algebra.Com's Answer #482042 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "There is obviously a trivial solution x=0, y=0,\r\n" );
document.write( "in which t can be any number.  So for any other\r\n" );
document.write( "case we will assume there are other soluitions.\r\n" );
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document.write( "For\r\n" );
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document.write( "4x+tx-20y = 0 \r\n" );
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document.write( "Factor x out of the first two terms on the left\r\n" );
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document.write( "x(4+t) - 20y = 0\r\n" );
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document.write( "(4+t)x - 20y = 0\r\n" );
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document.write( "For \r\n" );
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document.write( "x-4y+ty = 0 \r\n" );
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document.write( "Factor -y out of the last two terms on the left\r\n" );
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document.write( "x-y(4-t) = 0\r\n" );
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document.write( "x-(4-t)y = 0\r\n" );
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document.write( "So the system of equations is\r\n" );
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document.write( "(4+t)x -    20y = 0\r\n" );
document.write( "     x - (4-t)y = 0\r\n" );
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document.write( "Since both constant terms on the right are 0,\r\n" );
document.write( "that means the determinant of coefficient\r\n" );
document.write( "matrix must be 0, so \r\n" );
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document.write( "det(A) = 0\r\n" );
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document.write( "Evaluate the determinant on the left and we have\r\n" );
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document.write( "-(4+t)(4-t) - (-20)(1) = 0\r\n" );
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document.write( "-(16-t²) + 20 = 0\r\n" );
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document.write( "     -(16-t²) = -20\r\n" );
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document.write( "        16-t² = 20\r\n" );
document.write( "          -t² = 4\r\n" );
document.write( "           t² = -4\r\n" );
document.write( "            t = ±√-4\r\n" );
document.write( "            t = ±2i\r\n" );
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document.write( "Oh, oh!!!!!  Something is wrong with the problem!!!!!\r\n" );
document.write( "There are no real values of t!!!!!\r\n" );
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document.write( "You must have copied a sign or number wrong, for t \r\n" );
document.write( "cannot be a real number as the problem is given \r\n" );
document.write( "here, except for the one trivial solution x=0 and \r\n" );
document.write( "y=0, in which case t can be any number.\r\n" );
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document.write( "If you will state the problem correctly in the \r\n" );
document.write( "thank-you note form, we can help you.  Be sure to \r\n" );
document.write( "check the signs and numbers very carefully.  \r\n" );
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document.write( "Edwin

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