Cramer's rule
Apply Cramer's rule to solve systems of linear equations: compute determinants of coefficient and modified matrices, then divide to find each variable's value directly.
Key Concepts
The solution of the linear system $\begin{cases} ax + by = e \\ cx + dy = f \end{cases}$ is $x = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{D}$ and $y = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{D}$, where $D$ is the determinant of the coefficient matrix. This method uses determinants to solve systems of linear equations instead of elimination.
To solve $\begin{cases} 2x + 5y = 12 \\ x 2y = 3 \end{cases}$, we find $x = \frac{\begin{vmatrix} 12 & 5 \\ 3 & 2 \end{vmatrix}}{\begin{vmatrix} 2 & 5 \\ 1 & 2 \end{vmatrix}} = \frac{ 24 ( 15)}{ 4 5} = \frac{ 9}{ 9} = 1$; For the same system, we find $y = \frac{\begin{vmatrix} 2 & 12 \\ 1 & 3 \end{vmatrix}}{\begin{vmatrix} 2 & 5 \\ 1 & 2 \end{vmatrix}} = \frac{ 6 12}{ 4 5} = \frac{ 18}{ 9} = 2$.
Think of Cramer's rule as a fantastic shortcut for solving systems of equations, swapping out tricky algebra for simple arithmetic with determinants. The denominator determinant uses the variable coefficients and stays the same for both x and y. The numerator determinants are custom built: just replace the coefficients of the variable you're solving for with the constants from the equations.
Common Questions
What is Cramer's rule and when does it apply?
Cramer's rule is a formula for solving a system of n linear equations with n unknowns using determinants. It applies whenever the coefficient matrix A has a nonzero determinant. Each variable xi equals det(Ai) divided by det(A), where Ai is the coefficient matrix with the i-th column replaced by the constant column.
How do you solve a 2x2 system using Cramer's rule?
For the system ax+by=e and cx+dy=f, compute D=det([[a,b],[c,d]])=ad-bc. Then x=det([[e,b],[f,d]])/D=(ed-bf)/D and y=det([[a,e],[c,f]])/D=(af-ce)/D. If D=0 the system has no unique solution.
What is the advantage of Cramer's rule over substitution or elimination?
Cramer's rule gives a direct closed-form formula for each variable, which is useful when you need only one variable's value from a large system. However, it is computationally intensive for large systems. For small 2x2 or 3x3 systems it provides a systematic determinant-based approach.