Linear equations and determinants

$\displaystyle a_1x + b_1y = c_1\\\vspace{0.2 in} a_2x + b_2y = c_2\ \cdots(1)$
These represent two lines in the xy plane, and in general will meet in a point whose coordinates (x, y) are found by solving simultaneously.
$\displaystyle x = \frac{c_1b_2 - b_1c_2}{a_1b_2 - b_1a_2},\ y = \frac{a_1c_2 - c_1a_2}{a_1b_2 - b_1a_2}\ \cdots(2)$
It’s convenient to write these in determinant form as
$\displaystyle x = \frac{\left|\begin{array}{cc}c_1 & b_1 \\ c_2 & b_2\end{array}\right|}{\left|\begin{array}{cc}a_1 & b_1 \\ a_2 & b_2\end{array}\right|},\ y = \frac{\left|\begin{array}{cc}a_1 & c_1 \\ a_2 & c_2\end{array}\right|}{\left|\begin{array}{cc}a_1 & b_1 \\ a_2 & b_2 \end{array}\right|}\ \cdots(3)$
where it is defined a determinant of the second order or order 2 to be
$\displaystyle \left|\begin{array}{cc}a & b \\ c & d \end{array}\right| = ad - bc\ \cdots(4)$
It should be noted that the denominator for x and y in (3) is the determinant consisting of the coefficients of x and y in (1). The numerator for x is found by replacing the first column of the denominator by the constants c₁, c₂ on the right side of (1). Similarly the numerator for y is found by replacing the second column of the denominator by c₁, c₂. This procedure is often called Cramer’s rule. In case the denominator in (3) is zero, the two lines represented by (1) do not meet in one point but are either coincident or parallel.

The ideas are easily extended. Thus you can consider the equations
$\displaystyle a_1x + b_1y + c_1z = d_1\\\vspace{0.2 in} a_2x + b_2y + c_2z = d_2\ \cdots(5)\\\vspace{0.2 in} a_3x + b_3y + c_3z = d_3$
representing 3 planes. If they intersect in a point, the coordinates (x, y, x) of this point are found from Cramer’s rule to be
$\displaystyle x = \frac{\left|\begin{array}{ccc}d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3\end{array}\right|}{\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|},\ y = \frac{\left|\begin{array}{ccc}a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3\end{array}\right|}{\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|},\ z = \frac{\left|\begin{array}{ccc}a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3\end{array}\right|}{\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|}\ \cdots(6)$
where it can be defined the determinant of order 3 by
$\displaystyle \left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\a_3 & b_3 & c_3\end{array}\right| = a_1b_2c_3 + b_1c_2a_3 + c_1a_2b_3 - (b_1a_2c_3 + a_1c_2b_3 + c_1b_2a_3)\ \cdots(7)$
The determinant can also be evaluated in terms of second order determinants as follows
$\displaystyle a_1\left|\begin{array}{cc}b_2 & c_2 \\ b_3 & c_3\end{array}\right| - b_1\left|\begin{array}{cc}a_2 & c_2 \\ a_3 & c_3\end{array}\right| + c_1\left|\begin{array}{cc}a_2 & b_2 \\ a_3 & b_3\end{array}\right|\ \cdots(8)$
where it is noted that a₁, b₁, c₁ are the elements in the first row and the corresponding second order determinants are those obtained from the given third order determinant by removing the row and column in which the element appears.