Using determinants, solve the following equations simultaneously.
5x — 6y + 4z = 15
7x + 4y — 3z = 19
2x + y + 6z = 46
Explanation
To find the determinant of (begin{pmatrix} 5 & -6 & 4\ 7 & 4 & -3\ 2 & 1 & 6 end{pmatrix}) which is 5(24 + 3) + 6(4 + 6) + 4(7 – 8) = 135 + 288 – 4 = 419.
Then, substitude the column of the coefficient of x by the right – hand side of the equations and find the determinant of (begin{pmatrix} 5 & -6 & 4\ 7 & 4 & -3\ 46 & 1 & 6 end{pmatrix}) = 15(24 + 3) + 6(114 + 138) + 4(19 – 184) = 405 + 1512 – 660 = 1257.
Similarly, to find the coefficient of y, find the determinant of (begin{pmatrix} 5 & 15 & 4\ 7 & 19 & -3\ 2 & 46 & 6 end{pmatrix})
= 5(114 + 138) – 13(42 + 6) – 4(322 – 38) = 1260 – 720 + 1136 = 1676.
Finally, for the coefficient of z find the determinant of (begin{pmatrix} 5 & -6 & 15\ 7 & 19 & -3\ 2 & 46 & 6 end{pmatrix})
= 5(184 – 19) + 6(322 – 38) + 15(7 – 8) = 825 + 1704 – 15 = 2514
Therefore, x = (frac{1257}{419}) = 3, y = (frac{1676}{419}) = 4
and z = (frac{2514}{419}) = 6