document.write( "Question 934937: Solve each system\r
\n" ); document.write( "\n" ); document.write( "2x+y+z=14
\n" ); document.write( "-x-3y+2z=-2
\n" ); document.write( "4x-6y+3z=-5 \n" ); document.write( "
Algebra.Com's Answer #567987 by MathLover1(20850)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "\"2x%2By%2Bz=14\".....eq.1
\n" ); document.write( "\"-x-3y%2B2z=-2\".....eq.2
\n" ); document.write( "\"4x-6y%2B3z=-5\".....eq.3
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\n" ); document.write( "\n" ); document.write( "start with
\n" ); document.write( "\"2x%2By%2Bz=14\".....eq.1
\n" ); document.write( "\"-x-3y%2B2z=-2\".....eq.2...both sides multiply by \"2\"
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\n" ); document.write( "\n" ); document.write( "\"2x%2By%2Bz=14\".....eq.1
\n" ); document.write( "\"-2x-6y%2B4z=-4\".....eq.2
\n" ); document.write( "__________________________add\r
\n" ); document.write( "\n" ); document.write( "\"cross%282x%29%2By%2Bz-cross%282x%29-6y%2B4z=14-4\"\r
\n" ); document.write( "\n" ); document.write( "\"5z-5y=10\"\r
\n" ); document.write( "\n" ); document.write( "\"5%28z-y%29=10\"\r
\n" ); document.write( "\n" ); document.write( "\"z-y=10%2F5\"\r
\n" ); document.write( "\n" ); document.write( "\"z-y=2\"....solve for \"z\"\r
\n" ); document.write( "\n" ); document.write( "\"z=y%2B2\"...............1a\r
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\n" ); document.write( "\n" ); document.write( "\"-x-3y%2B2z=-2\".....eq.2...both sides multiply by \"4\"
\n" ); document.write( "\"4x-6y%2B3z=-5\".....eq.3
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\n" ); document.write( "\n" ); document.write( "\"-4x-12y%2B8z=-8\".....eq.2
\n" ); document.write( "\"4x-6y%2B3z=-5\".....eq.3
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\n" ); document.write( "\n" ); document.write( "\"-cross%284x%29-12y%2B8z%2Bcross%284x%29-6y%2B3z=-8-5\"\r
\n" ); document.write( "\n" ); document.write( "\"-18y%2B11z=-13\"...solve for \"z\"\r
\n" ); document.write( "\n" ); document.write( "\"11z=18y-13\"\r
\n" ); document.write( "\n" ); document.write( "\"z=18y%2F11-13%2F11\".............1b\r
\n" ); document.write( "\n" ); document.write( "1a and 1b have equal left sides, so right sides must be equal too\r
\n" ); document.write( "\n" ); document.write( "\"y%2B2=18y%2F11-13%2F11\".....solve for \"y\"\r
\n" ); document.write( "\n" ); document.write( "\"11y%2B22=18y-13\"\r
\n" ); document.write( "\n" ); document.write( "\"13%2B22=18y-11y\"\r
\n" ); document.write( "\n" ); document.write( "\"35=7y\"\r
\n" ); document.write( "\n" ); document.write( "\"35%2F7=y\"\r
\n" ); document.write( "\n" ); document.write( "\"highlight%28y=5%29\"\r
\n" ); document.write( "\n" ); document.write( "now find \"z\"\r
\n" ); document.write( "\n" ); document.write( "\"z=y%2B2\"...............1a\r
\n" ); document.write( "\n" ); document.write( "\"z=5%2B2\"\r
\n" ); document.write( "\n" ); document.write( "\"highlight%28z=7%29\"\r
\n" ); document.write( "\n" ); document.write( "go to one of the given equations, plug in values for \"y\" and \"z\" and find \"x\"\r
\n" ); document.write( "\n" ); document.write( "\"2x%2By%2Bz=14\".....eq.1\r
\n" ); document.write( "\n" ); document.write( "\"2x%2B5%2B7=14\"\r
\n" ); document.write( "\n" ); document.write( "\"2x%2B12=14\"\r
\n" ); document.write( "\n" ); document.write( "\"2x=14-12\"\r
\n" ); document.write( "\n" ); document.write( "\"2x=2\"\r
\n" ); document.write( "\n" ); document.write( "\"highlight%28x=1%29\"\r
\n" ); document.write( "\n" ); document.write( "so, your solutions are: \"highlight%28x=1%29\",\"highlight%28y=5%29\", and
\n" ); document.write( "\"highlight%28z=7%29\"\r
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\n" ); document.write( "\n" ); document.write( "this is another way to solve this system:\r
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Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables

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\n" ); document.write( " First let \"A=%28matrix%283%2C3%2C2%2C1%2C1%2C-1%2C-3%2C2%2C4%2C-6%2C3%29%29\". This is the matrix formed by the coefficients of the given system of equations.
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\n" ); document.write( " Take note that the right hand values of the system are \"14\", \"-2\", and \"-5\" and they are highlighted here:
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\n" ); document.write( " These values are important as they will be used to replace the columns of the matrix A.
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\n" ); document.write( " Now let's calculate the the determinant of the matrix A to get \"abs%28A%29=35\". To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.
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\n" ); document.write( " Notation note: \"abs%28A%29\" denotes the determinant of the matrix A.
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\n" ); document.write( " Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix \"A%5Bx%5D\" (since we're replacing the 'x' column so to speak).
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\n" ); document.write( " Now compute the determinant of \"A%5Bx%5D\" to get \"abs%28A%5Bx%5D%29=35\". Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.
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\n" ); document.write( " To find the first solution, simply divide the determinant of \"A%5Bx%5D\" by the determinant of \"A\" to get: \"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%2835%29%2F%2835%29=1\"
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\n" ); document.write( " So the first solution is \"x=1\"
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\n" ); document.write( " We'll follow the same basic idea to find the other two solutions. Let's reset by letting \"A=%28matrix%283%2C3%2C2%2C1%2C1%2C-1%2C-3%2C2%2C4%2C-6%2C3%29%29\" again (this is the coefficient matrix).
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\n" ); document.write( " Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix \"A%5By%5D\" (since we're replacing the 'y' column in a way).
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\n" ); document.write( " Now compute the determinant of \"A%5By%5D\" to get \"abs%28A%5By%5D%29=175\".
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\n" ); document.write( " To find the second solution, divide the determinant of \"A%5By%5D\" by the determinant of \"A\" to get: \"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28175%29%2F%2835%29=5\"
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\n" ); document.write( " So the second solution is \"y=5\"
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\n" ); document.write( " Let's reset again by letting \"A=%28matrix%283%2C3%2C2%2C1%2C1%2C-1%2C-3%2C2%2C4%2C-6%2C3%29%29\" which is the coefficient matrix.
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\n" ); document.write( " Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix \"A%5Bz%5D\"
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\n" ); document.write( " Now compute the determinant of \"A%5Bz%5D\" to get \"abs%28A%5Bz%5D%29=245\".
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\n" ); document.write( " To find the third solution, divide the determinant of \"A%5Bz%5D\" by the determinant of \"A\" to get: \"z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%28245%29%2F%2835%29=7\"
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\n" ); document.write( " So the third solution is \"z=7\"
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\n" ); document.write( " Final Answer:
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\n" ); document.write( " So the three solutions are \"x=1\", \"y=5\", and \"z=7\" giving the ordered triple (1, 5, 7)
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\n" ); document.write( " Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.
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