Using determinants, solve the following equations simultaneously. 5x — 6y + 4z = 15 7x
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