document.write( "Question 25009: show that if Ax=b has more than one solution, then it has infinitely many solutions(Hint: If x1 and x2 are solutions, consider x3=rx1 +sx2, where r+s=1) \n" ); document.write( "

Algebra.Com's Answer #13414 by venugopalramana(3286) $\"\"$ $\"About$

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show that if Ax=b has more than one solution, then it has infinitely many solutions(Hint: If x1 and x2 are solutions, consider x3=rx1 +sx2, where r+s=1)
\n" ); document.write( "AX=B
\n" ); document.write( "LET THE 2 SOLUTIONS BE X1 AND X2.HENCE
\n" ); document.write( "AX1=B.....................I... AND
\n" ); document.write( "AX2=B.....................II
\n" ); document.write( "MULTIPLYING X1 WITH A CONSTANT r,WE GET FROM EQN.I
\n" ); document.write( "A*rX1=r*AX1=rB.............III
\n" ); document.write( "SIMILARLY MULTIPLYING X2 WITH A CONSTANT s,WE GET FROM EQN.II
\n" ); document.write( "A*sX2=s*AX2=sB.............IV
\n" ); document.write( "ADDING EQNS.III AND IV,WE GET
\n" ); document.write( "A*(rX1+sX2)=(r+s)B
\n" ); document.write( "IF WE TAKE THE 2 CONSTANTS r AND s SUCH THAT r+s=1,WE HAVE
\n" ); document.write( "A*(rX1+sX2)=1B=B
\n" ); document.write( "HENCE (rX1+sX2) IS ALSO A SOLUTION.THUS BY TAKING DIFFERENT VALUES FOR r AND SUCH THAT r+s=1,WE GET INFINITE SOLUTIONS. \n" ); document.write( "

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