We are now introducing a method for solving systems of equations that uses determinants, known as Cramer’s Rule. This technique
date back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752),
who introduced it in 1750 in Introduction to the analysis of lines algebraic curves. Cramer’s Rule is a viable and efficient method for
finding solutions to systems with an arbitrary number of unknowns, if we have the same number of equations as unknowns.
Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an
infinite number of solutions, this will be indicate by a determinant of zero. To find out if the system is inconsistent or dependent,
another method, such as elimination, will have to be use.

Using Cramer’s Rule to Solve a System of Two Equations in

Two Variables

To understand Cramer’s Rule, let us look closely at how we solve
systems of linear equations using basic row operations. Consider a
system of two equations in two variables.
a1x + b1y = c1 → equation 01
a2x + b2y = c2 → equation 02

We eliminate one variable using row operations and solve for the other.
Say that we wish to solve for x. If equation 02 is multiply by the opposite
of the coefficient of y in equation 01, equation 01 is multiply by the
coefficient of y in equation 02, and we add the two equations, the variable
y will be eliminating.

b2a1x + b2b1y = b2c1 Multiply R1 by b2

-b1a2x – b1b2y = -b1c2 Multiply R2 by –b2

b2a1x – b1a2x = b2c1 – b1c2

Now, solve for x.

b2a1x – b1a2x = b2c1 – b1c2
x (b2a1 – b1a2) = b2c1 – b1c2

x =


[c1 b1 c2 b2]
[a1 b1a2 b2]
Similarly to solve for y, we will eliminate x.

a2a1x + a2b1y = a2c1 Multiply R1 by a2

-a2a1x – a1b2y = -a1c2 Multiply R2 by –a1

a2b1y – a1b2y = a2c1 – a1c2

Solving for y gives

a2b1y – a1b2y = a2c1 – a1c2
y (a2b1 – a1b2) = a2c1 – a1c2

y =




[a1 c1 a2 c2]
[a1 b1 a2 b2]

Notice that the denominator for x and y is the determinant of the
coefficient matrix.


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