document.write( "Question 1151846: 7x − y − z = 34
\n" ); document.write( "x − 3y + z = 6
\n" ); document.write( "x + 2y − z = 4
\n" ); document.write( "This is my first question! Find, x,y,z and make sure that they are all true!\r
\n" ); document.write( "\n" ); document.write( "4a − 3b = 1
\n" ); document.write( "6a − 8c = 1
\n" ); document.write( "2b − 4c = 0
\n" ); document.write( "This is my second question! find, x,y,z and make sure that they are all true! \n" ); document.write( "
Algebra.Com's Answer #773724 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "I'll do the first problem to get you started\r
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\n" ); document.write( "\n" ); document.write( "Given system of equations
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document.write( "7x − y − z = 34 ..... equation (1)\r\n" );
document.write( "x − 3y + z = 6  ..... equation (2)\r\n" );
document.write( "x + 2y − z = 4  ..... equation (3)\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solve equation (1) for z
\n" ); document.write( "7x - y - z = 34
\n" ); document.write( "7x - y - z+z = 34+z
\n" ); document.write( "7x - y = 34+z
\n" ); document.write( "7x - y - 34 = 34+z - 34
\n" ); document.write( "7x - y - 34 = z
\n" ); document.write( "z = 7x - y - 34 .... call this equation (4)\r
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\n" ); document.write( "\n" ); document.write( "Plug this into equation (2). Simplify. Then solve for y.
\n" ); document.write( "x - 3y + z = 6
\n" ); document.write( "x - 3y + 7x - y - 34 = 6
\n" ); document.write( "8x - 4y - 34 = 6
\n" ); document.write( "8x - 4y = 40
\n" ); document.write( "2x - y = 10
\n" ); document.write( "y = 2x - 10 ... we'll use this later.\r
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\n" ); document.write( "\n" ); document.write( "Move onto equation (3) and plug in z = 7x - y - 34. Simplify
\n" ); document.write( "x + 2y - z = 4
\n" ); document.write( "x + 2y - (7x - y - 34) = 4
\n" ); document.write( "x + 2y - 7x + y + 34 = 4
\n" ); document.write( "-6x + 3y + 34 = 4
\n" ); document.write( "-6x + 3y = -30
\n" ); document.write( "-2x + y = -10\r
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\n" ); document.write( "\n" ); document.write( "Next replace y with 2x-10, which was from y = 2x-10 found earlier
\n" ); document.write( "-2x + y = -10
\n" ); document.write( "-2x + 2x - 10 = -10
\n" ); document.write( "-10 = -10
\n" ); document.write( "Both sides are the same value.
\n" ); document.write( "We have a consistent and dependent system.
\n" ); document.write( "There are infinitely many solutions\r
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\n" ); document.write( "\n" ); document.write( "Go back to equation (4). Plug in y = 2x-10
\n" ); document.write( "z = 7x - y - 34
\n" ); document.write( "z = 7x - (2x - 10) - 34
\n" ); document.write( "z = 7x - 2x + 10 - 34
\n" ); document.write( "z = 5x - 24
\n" ); document.write( "We have now expressed z in terms of x\r
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\n" ); document.write( "\n" ); document.write( "The variable y is in terms of x as well
\n" ); document.write( "y = 2x - 10\r
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\n" ); document.write( "\n" ); document.write( "We can let x be the free variable. It is anything you want while y and z depend on what x is
\n" ); document.write( "x = any number
\n" ); document.write( "y = 2x-10
\n" ); document.write( "z = 5x-24\r
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\n" ); document.write( "\n" ); document.write( "One way to express all the solutions is to write
\n" ); document.write( "(x, y, z) = (x, 2x-10, 5x-24)
\n" ); document.write( "This x,y,z triple will allow you to generate any solution you want\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "For instance, if x = 0, then
\n" ); document.write( "(x, y, z) = (x, 2x-10, 5x-24)
\n" ); document.write( "(x, y, z) = (0, 2*0-10, 5*0-24)
\n" ); document.write( "(x, y, z) = (0, -10, -24)
\n" ); document.write( "which is one solution\r
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\n" ); document.write( "\n" ); document.write( "Or if x = 7, then
\n" ); document.write( "(x, y, z) = (x, 2x-10, 5x-24)
\n" ); document.write( "(x, y, z) = (7, 2*7-10, 5*7-24)
\n" ); document.write( "(x, y, z) = (7, 4, 11)
\n" ); document.write( "which is another solution. \r
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\n" ); document.write( "\n" ); document.write( "Or if x = 5, then
\n" ); document.write( "(x, y, z) = (x, 2x-10, 5x-24)
\n" ); document.write( "(x, y, z) = (5, 2*5-10, 5*5-24)
\n" ); document.write( "(x, y, z) = (5, 0, 1)
\n" ); document.write( "which is another solution. \r
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\n" ); document.write( "\n" ); document.write( "This process goes on forever because x can be any real number (and there are infinitely many of those).\r
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\n" ); document.write( "\n" ); document.write( "note: x does not have to be the free variable. We could make either y or z have that role. I would ask your teacher which s/he prefers to have as the free variable. Often times, it is sufficient enough to say \"infinitely many solutions\" and not have to specify the solution form.
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