document.write( "Question 1111109: Solve the equations by using elimination method and Use determinants to solve systems of linear equations
\n" ); document.write( "2x+y+3z=-9
\n" ); document.write( "3x-2y+5z=-8
\n" ); document.write( "x+7y-4z=-1 \n" ); document.write( "
Algebra.Com's Answer #726129 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
to solve by determinants, the general concept is:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "first create a matrix with the coefficients and the constants of the system of equations.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the system of equations given is:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "          2x     +     y     +    3z     =     -9\r\n" );
document.write( "          3x     -    2y     +    5z     =     -8\r\n" );
document.write( "           x     +    7y     -    4z     =     -1\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "your matrix will look like this:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "          \r\n" );
document.write( "           2     1     3   -9\r\n" );
document.write( "           3    -2     5   -8\r\n" );
document.write( "           1     7    -4   -1\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the first column corresponds to the coefficients of the x variable.
\n" ); document.write( "the second column corresponds to the coefficients of the y variable.
\n" ); document.write( "the third column corresponds to the coefficients of the z variable.
\n" ); document.write( "the fourth column corresponds to the constants.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "you first find the determinants of the 3 by 3 matrix formed from the first 3 columns.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "that matrix will look like this:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "           \r\n" );
document.write( "           2     1     3   \r\n" );
document.write( "           3    -2     5   \r\n" );
document.write( "           1     7    -4   \r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the first column corresponds to the coefficients of the x variable.
\n" ); document.write( "the second column corresponds to the coefficients of the y variable.
\n" ); document.write( "the third column corresponds to the coefficients of the z variable. \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "that determinant will be equal to 32, which we'll call D.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "you then replace the first column with the fourth column and find the determinant of that 3 by 3 matrix.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "that matrix will look like this:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "           \r\n" );
document.write( "          -9     1     3   \r\n" );
document.write( "          -8    -2     5   \r\n" );
document.write( "          -1     7    -4   \r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the first column corresponds to the constants.
\n" ); document.write( "the second column corresponds to the coefficients of the y variable.
\n" ); document.write( "the third column corresponds to the coefficients of the z variable. \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "that determinant will be equal to 32, which we'll call Dx.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "you then go back to the original matrix and replace the second column with the fourth column and find the determinant of of that 3 by 3 matrix.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "that matrix will look like this.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "          \r\n" );
document.write( "           2    -9     3   \r\n" );
document.write( "           3    -8     5   \r\n" );
document.write( "           1    -1    -4   \r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the first column corresponds to the coefficients of the x variable.
\n" ); document.write( "the second column corresponds to the constants.
\n" ); document.write( "the third column corresponds to the coefficients of the z variable. \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "that determinant will be equal to -64, which we'll call Dy.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "you then go back to the original matrix and replace the third column with the fourth column and find the determinant of that 3 by 3 matrix.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "that matrix will look like this.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "           x     y     c\r\n" );
document.write( "           2     1    -9\r\n" );
document.write( "           3    -2    -8\r\n" );
document.write( "           1     7    -1\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the first column corresponds to the coefficients of the x variable.
\n" ); document.write( "the second column corresponds to the coefficients of the y variable.
\n" ); document.write( "the third column corresponds to the constants.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "that determinant will be equal to -96, which we'll call Dz.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the value of x is equal to Dx / D which is equal to 32/32 = 1.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the value of y is equal to Dy / D whcih is equal to -64/32 = -2.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the value of z is equal to Dz / d which is equal to -96/32 = -3.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the solution to your problem is:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "x = 1
\n" ); document.write( "y = -2
\n" ); document.write( "z = -3\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "i solve this problem using online calculators and got the same results so i'm reasonably confident the answer is correct.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "here's a reference on how to calculate a determinant.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://www.mathsisfun.com/algebra/matrix-determinant.html\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "here's a reference on how to use determinants to solve a system of linear equations.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://www.youtube.com/watch?v=TtxVGMWXMSE\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );