document.write( "Question 44237This question is from textbook College Algebra
\n" ); document.write( ": Solve each system by substitution. Determine whether each system is independent, inconsistent, or dependent\r
\n" ); document.write( "\n" ); document.write( "2y = 1 - 4x
\n" ); document.write( "2x + y = 0 \n" ); document.write( "
Algebra.Com's Answer #29171 by AnlytcPhil(1807)\"\" \"About 
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Solve each system by substitution. Determine whether each system is \r\n" );
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document.write( "(a)\r\n" );
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document.write( " x + 2y = 3\r\n" );
document.write( "2x +  y = 5	\r\n" );
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document.write( "Solve the first equation for x\r\n" );
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document.write( "        x + 2y = 3\r\n" );
document.write( "             x = 3 - 2y\r\n" );
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document.write( "Substitute (3 - 2y) for x in the second equation:\r\n" );
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document.write( "        2x + y = 5\r\n" );
document.write( " 2(3 - 2y) + y = 5\r\n" );
document.write( "    6 - 4y + y = 5\r\n" );
document.write( "        6 - 3y = 5\r\n" );
document.write( "           -3y = 5 - 6\r\n" );
document.write( "           -3y = -1\r\n" );
document.write( "             y = 1/3 \r\n" );
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document.write( "Now substitute 1/3 for y in     	\r\n" );
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document.write( "             x = 3 - 2y\r\n" );
document.write( "             x = 3 - 2(1/3)\r\n" );
document.write( "             x = 3 - 2/3\r\n" );
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document.write( "Clear of fractions by multiplying thru by LCD = 3\r\n" );
document.write( "            3x = 9 - 2\r\n" );
document.write( "            3x = 7\r\n" );
document.write( "             x = 7/3\r\n" );
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document.write( "This is the most common type of system, when there is exactly\r\n" );
document.write( "one solution.  If you graphed the lines they would cross in \r\n" );
document.write( "one point (7/3, 1/3).  It is called an independent (consistent) \r\n" );
document.write( "system.\r\n" );
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document.write( "(b)\r\n" );
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document.write( "2y = 1 - 4x\r\n" );
document.write( "2x + y = 0\r\n" );
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document.write( "Solve the second equation for y\r\n" );
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document.write( "        2x + y = 0\r\n" );
document.write( "             y = -2x\r\n" );
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document.write( "Substitute (-2x) for y in the first equation:\r\n" );
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document.write( "            2y = 1 - 4x\r\n" );
document.write( "        2(-2x) = 1 - 4x\r\n" );
document.write( "           -4x = 1 - 4x\r\n" );
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document.write( "Add 4x to both sides:\r\n" );
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document.write( "             0 = 1\r\n" );
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document.write( "Uh! oh!  Zero just doesn't equal to one! \r\n" );
document.write( "We can't go any further because there are\r\n" );
document.write( "no variables or unknowns in the equation. But\r\n" );
document.write( "since we didn't do anything wrong, if we \r\n" );
document.write( "believed there is any solution to this system, \r\n" );
document.write( "we'd have to also believe that zero equals one!\r\n" );
document.write( "But we don't believe that.  So we must not\r\n" );
document.write( "believe this system has a solution either! It\r\n" );
document.write( "doesn't.\r\n" );
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document.write( "This is a special type of system, when there is \r\n" );
document.write( "no solution.  This is called an inconsistent\r\n" );
document.write( "system.  \r\n" );
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document.write( "-----\r\n" );
document.write( "The third type -- you don't have one here -- is when you \r\n" );
document.write( "get something like 0 = 0  or -1 = -1, when you can't go \r\n" );
document.write( "any further but the equation is just \"a number = itself\", \r\n" );
document.write( "which is true, so the system can't be inconsistent, so it \r\n" );
document.write( "has infinitely many solutions, and is called a dependent \r\n" );
document.write( "system.\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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