document.write( "Question 1062149: Write a matrix for a system of equations that does not
\n" ); document.write( "have a unique solution. Why is the solution not unique?
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #677030 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!
Write a matrix for a system of equations that does not
\n" ); document.write( "have a unique solution.
\n" ); document.write( "

\r\n" );
document.write( "Make up an easy equation that has more than one solution:\r\n" );
document.write( "\r\n" );
document.write( "x + y = 10\r\n" );
document.write( "\r\n" );
document.write( "That has lots of solutions because \r\n" );
document.write( "\r\n" );
document.write( "1 + 9 = 10, 2 + 8 = 10, 5 + 5 = 10, 6 + 4 = 10, etc., etc.\r\n" );
document.write( "\r\n" );
document.write( "Now multiply that equation through by some easy number, say, -1:\r\n" );
document.write( "\r\n" );
document.write( "-x - y = -10\r\n" );
document.write( "\r\n" );
document.write( "That one has the same solutions that the first one does.\r\n" );
document.write( "\r\n" );
document.write( "So this system of equation has many solutions:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now make it into an augmented matrix\r\n" );
document.write( "\r\n" );
document.write( "

Why is the solution not unique?

Answer:  Because we picked an equation that has many solutions.\r\n" );
document.write( "Then we multiplied it through to get another equation that \r\n" );
document.write( "has the same solutions.\r\n" );
document.write( "\r\n" );
document.write( "Also if we add row 1 to row 2, we get the\r\n" );
document.write( "row-reduced echelon form:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "And we have learned that when the augmented matrix is\r\n" );
document.write( "in row reduced echelon form and the bottom row has all\r\n" );
document.write( "0s, then the solution is not unique. The matrix is\r\n" );
document.write( "equivalent to the equation:\r\n" );
document.write( "\r\n" );
document.write( "x + y = 10\r\n" );
document.write( "\r\n" );
document.write( "and solving for x we get\r\n" );
document.write( "\r\n" );
document.write( "x = 10 - y\r\n" );
document.write( "\r\n" );
document.write( "So the general solution is\r\n" );
document.write( "\r\n" );
document.write( "(10-y, y)\r\n" );
document.write( "\r\n" );
document.write( "We can substitute any number for y and get a different\r\n" );
document.write( "solution.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );