Cramer’s Rule

Cramer’s rule is a method to solve linear equations quickly. In the system of linear equations, two unknowns can be solved by various methods that include (1) graphing, (2) elimination, and (3) substitution. You might ask me, which method do you prefer to use?” My preference is to use none of the methods we have discussed—in favor of a very easy approach that is presented called Cramer’s Rule and which involves the use of matrices (plural form of the word “matrix”). I will share with you my “secret” method, but you must utilize this method after you understand this particular approach taken an upper-level math course.
Let’s consider the system of linear equations that we solved earlier when we were graphing the solutions,

y = -2x + 3
y = x – 6

We are going to rewrite these equations in standard form, rather than work with the equations in slope and y-intercept format. The equivalent system of equations in standard form is

2x + y = 3
-x + y = -6

Rewriting these equations showing the implied coefficients, we have

2x + 1y = 3
-1x + 1y = -6

Next, we can form a matrix of the coefficients of x and y as shown,


Next, we compute what is called the determinant of the coefficient matrix by multiplying the number in the top left of the matrix times the number in the bottom right and then subtracting the product of the number in the bottom left of the matrix and the number in the upper right. Thus, we have the determinant computed as follows:

D= $$\begin{vmatrix}2&1\-1&1\end{vmatrix}$$ = (2)(1) – (-1)(1) = 2 + 1 = 3

We will also need to compute Dx and Dy. So based on the general system of equations:

a1x + b1y = c1
a2x + b2y = c2

The determinants are computed as follows:

D= $$\begin{bmatrix}a1&b1\a2&b2\end{bmatrix}$$ = a1b2 – a2b1

Dx = $$\begin{bmatrix}c1&b1\c2&b2\end{bmatrix}$$ = = c1b2 – c2b

Dy = $$\begin{bmatrix}a1&c1\a2&c2\end{bmatrix}$$ = = a1c2 – a2 c1

Then, the values of x and y that satisfy the equations are simply given by:

x = Dx/D
y = Dy/D

In summary, the x and y solutions can be computed for any system of two equations in two unknowns using these equivalent formulas (derived using Cramer’s Rule):

x = (c1b2 – c2b1)/(a1b2 – a2b1) or x = (c1b2 – c2b1)/D

y = (c2a1 – c1a2)/(a1b2 – a2b1) or y = (c2a1 – c1a2)/D

The solution to our system of linear equations is:

x = [(3)(1) – (-6)(1)]/3 = (3 + 6)/3 = 9/3 = 3
y = [(-6)(2) – (3)(-1)]/3 = (-12 + 3)/3 = -9/3 = -3

Thus, the solution is (3,-3).

It should be noted that if the determinant of the coefficient matrix is 0, then either the lines are parallel or they are identical. What is interesting about Cramer’s rule is that it can be used for systems of equations involving more than two unknowns.


Given the system of equations in two unknowns
a1x + b1y = c1
a2x + b2y = c2
The coefficient matrix determinant isD = a1b2 – a2b1
The solution for x isx = (c1b2 – c2b1)/D
The solution for y isy = (c2a1 – c1a2)/D
Cramer’s Rule

Cramer’s rule

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