That is also known as "Cramer's rule" which I have always considered to be much too "formulaic", not allowing "thinking" about any special properties of the equations that might simplify the solution.

MUHAMMED ADEBAYO KASSIM PICKING METHODS (MAK METHODS)
PICK DIRECT METHODS
TWO EQUATION TWO UNKNOWNS

X + 3Y = 5
X - Y = - 1
X + 3Y - 5= 0
X - Y + 11 = 0

Pick the first two column coefficient,ie first and second. Pick the next two column coefficients ie second and third and Pick the next two column coefficients ie third and first.
∆ = |1 3| = -1 -3 = -4
|1 -1|
∆1= |3 -5| = 33 -5 = 28
|- 1 11|
∆2 = | -5 1| = -5 – 11= -16
|11 1|
X = ∆1 = 28 = -7
∆ -4
Y = ∆2 = -16 = 4
∆ - 4




THREE EQUATIONS THREE UNKNOWNS

X - Y + Z = 12
2X -3Y – 2Z = 7
X + Y + Z = 6

X - Y + Z - 12 = 0
2X - 3Y - 2Z -7= 0
X + Y + Z - 6 = 0
Pick the first three column coefficients ie first, second and third. Leave the first column coefficient and pick the next three column coefficients ie second, third and first. Leave the second column coefficient and pick the next three column coefficient ie third, fourth and first. Leave the third column coefficient and pick the next three column coefficient ie fourth, first and second.

∆= |1 -1 1|
|2 -3 -2|
|1 1 1|
= 1| -3 - 2| - (-1) | 2 -2| + 1| 2 -3|
|1 1| |1 1| |1 1|
= 1( -3 + 2) + 1( 2 + 2) + 1 ( 2 + 3)
= 1( -1) + 1(4) + 1(5)
= - 1 + 4 + 5
= 8



∆1 = | -1 1 -12|
|-3 -2 -7 |
| 1 1 -6 |
= - 1| -2 - 7| - 1 | -3 -7| -12 | -3 -2|
| 1 -6 | | 1 -6| | 1 1|
= - 1 ( 12 + 7) - 1( 18 + 7) – 12(- 3 + 2)
= - 1( 19) – 1( 25) – 12(-1)
= - 19 – 25 + 12
= -32

∆2 = | 1 - 12 1|
| -2 -7 2|
| 1 - 6 1|
= 1| -7 2| - ( -12) | -2 2| + 1 | -2 -7|
|-6 1| |1 1| |1 -6|
= 1( -7 + 12) + 12( -2 -2) + 1 (12 + 7)
= 1(5) + 12( -4) + 1( 19)
= 5 -48 + 19
= -24

∆3= | -12 1 - 1|
| -7 2 -3|
| -6 1 1|
= -12| 2 -3| - 1 | -7 -3| -1 | -7 2|
|1 1| | -6 1| |-6 1|
= -12( 2 + 3) – 1( -7 -18) – 1( -7 + 12)
= - 12(5) – 1( -25) -1( -5)
= -60 + 25 – 5
= -40

X = -∆1 = -( -32) = 32 = 4
∆ 8 8
Y= + ∆2 = -24 = -3
∆ 8
Z = - ∆3 = - ( -40) = 5
∆ 8





Generally for two, four, six unknowns etc ( even values) has a positive sign at the front of their solutions while three, five, seven unknowns have a negative – positive order sign in front of their solutions.
Two unknowns, X1= ∆1 X2 = ∆2
∆ ∆
Four unknowns, X1 = - ∆1 X2 = ∆2 X3 = - ∆3
∆ ∆ ∆
Five unknowns, X1 = -∆1 X2 = ∆2 X3 = - ∆3 X4 = ∆4 X5 = -∆5
∆ ∆ ∆ ∆ ∆

References
Kassim Adebayo, 2019, Matrices Picking Methods- Faster than you ever thought possible https://www.amazon.com/gp/aw/d/B087637F ... p_img?is=l