Using Cramer’s Rule, find the solution to the following system of linear equations. 2x+3y=1, −x+y=−3

Question

Equations

2x+3y=1

-x+y=-3

Answer 1

Cramer's rule uses a set of determinants to calculate the values of "x" and "y" like this:

Given that a set of equations is in the form:

a₁x + b₁y = c₁

a₂x + b₂y = c₂

this set of equations has determinants defined as:

(Notice how D is the determinant of the coefficients, D_x is that same determinant except the "x" coefficients were replaced with the constant terms, and Dy is that same determinant except the "y" coefficients were replaced with the constant terms.)

Then x = D_x/D and y = D_y/D

To calculate the values of each determinant, you multiply the upper left and lower right numbers then subtract the product of the upper right and lower left. So:

D = (2)(1) - (3)(-1) = 2 - -3 = 5

D_x = (1)(1) - (3)(-3) = 1 - -9 = 10

D_y = (2)(-3) - (1)(-1) = -6 - -1 = -5

So x = 10/5 = 2

and y = -5/5 = -1