document.write( "Question 241047: The sum of 3 numbers is 26. The second number is three times the first while the third number is four more than the first. What are the three numbers?\r
\n" ); document.write( "\n" ); document.write( "I just need help setting it up. I can use Cramer's rule well enough, I'm just not sure what to put into it to solve it. \n" ); document.write( "

Algebra.Com's Answer #176539 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let x = the first number
\n" ); document.write( "Let y = the second number
\n" ); document.write( "Let z = the third number

\n" ); document.write( "The sum of 3 numbers is 26.
\n" ); document.write( "x + y + z = 26
\n" ); document.write( "The second number is three times the first
\n" ); document.write( "y = 3x
\n" ); document.write( "while the third number is four more than the first
\n" ); document.write( "z = x + 4

\n" ); document.write( "So the system is
\n" ); document.write( "x + y + z = 26
\n" ); document.write( "y = 3x
\n" ); document.write( "z = x + 4

\n" ); document.write( "To use Cramer's Rule we need to have all the variables, in order, on the left side and the constant terms on the right side. So we'll add -3x to both sides of the second equation and -x to both sides of the third equation:
\n" ); document.write( "x + y + z = 26
\n" ); document.write( "-3x + y = 0
\n" ); document.write( "-x + z = 4

\n" ); document.write( "Now we can use Cramer's Rule. We set up three fractions, one for each variable, where the numerator and denominator of each fraction are determinants.

The denominator of all three fractions is the determinant of the coefficient matrix (using zeros for missing variables).
The numerator of each fraction is the same determinant as the denominator except a column is replaced with the column matrix of the constant terms. The column of coefficients that gets replaced is the column of coefficients for the particular variable for which the fraction is the solution. (If this is not clear, look at the numerator and denominator of each fraction below. Note the difference between the two. Note where the column of constant terms, starting with 26, ends up.

\n" ); document.write( "I've set up the determinants below. (Algebra.com's software does not display determinants well. So the display is a bit crude.)
\n" ); document.write( "

\r\n" );
document.write( "         |  26   1   1   |\r\n" );
document.write( "         |   0   1   0   |\r\n" );
document.write( "         |   4   0   1   |\r\n" );
document.write( "x =    ---------------------\r\n" );
document.write( "         |   1   1   1   |\r\n" );
document.write( "         |  -3   1   0   |\r\n" );
document.write( "         |  -1   0   1   |\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         |   1  26   1   |\r\n" );
document.write( "         |  -3   0   0   |\r\n" );
document.write( "         |  -4   4   1   |\r\n" );
document.write( "y =    ---------------------\r\n" );
document.write( "         |   1   1   1   |\r\n" );
document.write( "         |  -3   1   0   |\r\n" );
document.write( "         |  -1   0   1   |\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         |   1   1  26   |\r\n" );
document.write( "         |  -3   1   0   |\r\n" );
document.write( "         |  -1   0   4   |\r\n" );
document.write( "z =    ---------------------\r\n" );
document.write( "         |   1   1   1   |\r\n" );
document.write( "         |  -3   1   0   |\r\n" );
document.write( "         |  -1   0   1   |\r\n" );
document.write( "\r\n" );
document.write( "

\n" ); document.write( "Now all that is left is to calculate the four determinants and simplify/reduce the fractions. \n" ); document.write( "

\n" );