When solving simultaneous equations (or systems of equations), several methods can be used such as the substitution method or the elimination method. Another system Cramer's Rule, named after the Swiss mathematician Gabriel Cramer (1704-1752) involves placing the coefficients and constants of each equation into determinants.|
Solving For Two Unknowns
If we have two equations:
ax + by = e
cx + dy = f
we can use Cramer's Rule by putting the 4 coefficients (a, b, c and d) and 2 constants (e&f) into 3 second order determinants.
(A second order determinant has 4 numbers arranged in 2 columns by 2 rows.)
The denominator determinant (dn) is created from the coefficients on the left side of the equations.
The x determinant's numerator resembles the dn determinant except the 'x' coefficients (a & c) are replaced by the constants (e & f).
The y determinant's numerator resembles the dn determinant except the 'y' coefficients (b & y) are replaced by the constants (e & f).
If we were to evaluate these determinants, it would be done based on this procedure:
Now that we know Cramer's Rule for 2 unknowns and evaluating determinants, let's solve some equations.
2x + 3y = 12
3x - 4y = 1
From the above instructions, we can see that:
a = 2 b = 3 c =3 d = -4 e = 12 f = 1
dn = (a d) - (c b)
dn = (2 -4) - (3 3)
dn = -17
x = [(e d) - (f b)] ÷ dn
x = [(12 -4) - (1 3)] ÷ -17
x = 3
y = [(a f) -(c e)] ÷ dn
y = [(2 1) -(3 12)] ÷ -17
y = 2
Cramer's Rule determines the solutions in the two unknown calculator.