Lesson 16: Using Cramer's Rule

New Concept

Cramer's rule is a method for solving systems of linear equations using determinants.

What’s next

Next, you’ll learn the specific formulas for $x$ and $y$ and apply them to solve systems of equations.

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.

The elements of the determinants in the denominators are the coefficients of $x$ and $y$ in the given equations. For this reason, this matrix is called the coefficient matrix. The order of the elements in the coefficient matrix is always the same. For the system $\begin{cases} ax + by = e \\ cx + dy = f \end{cases}$, the determinant is $D = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$.

For the system $\begin{cases} 3x + 2y = -1 \\ 4x - 3y = 10 \end{cases}$, the coefficient matrix determinant is $D = \begin{vmatrix} 3 & 2 \\ 4 & -3 \end{vmatrix} = -9 - 8 = -17$; For the system $\begin{cases} 6C + 6H = 78 \\ 10C + 8H = 128 \end{cases}$, the coefficient matrix determinant is $D = \begin{vmatrix} 6 & 6 \\ 10 & 8 \end{vmatrix} = 48 - 60 = -12$.

This is your system's unique fingerprint, built from the coefficients in front of your variables. You must keep them in the exact order they appear in the equations! This base determinant, called D, is the essential denominator for both the x and y solutions in Cramer's rule. Getting this matrix right is the crucial first step to finding the correct answer.

If the determinant of the coefficient matrix $D = 0$, the expression for the solution is undefined. If $D = 0$ but neither of the numerators is zero, the system has no solution. If $D = 0$ and at least one of the numerators is also zero, the system has an infinite number of solutions because the equations represent the same line.

For $\begin{cases} 2x + 4y = 7 \\ 2x + 4y = 9 \end{cases}$, $D=0$ and the numerator for $x$ is $\begin{vmatrix} 7 & 4 \\ 9 & 4 \end{vmatrix} = -8 \neq 0$. This system has no solution.; For $\begin{cases} x + 2y = -4 \\ 3x + 6y = -12 \end{cases}$, $D=0$ and the numerator for $x$ is $\begin{vmatrix} -4 & 2 \\ -12 & 6 \end{vmatrix} = 0$. This system has infinite solutions.

A zero in the denominator (D=0) signals special cases! It means the lines are either parallel and never meet, resulting in no solution, or they're secretly the same line, giving infinite solutions. To find out which, check the numerators. If the numerator is non-zero, it's a dead end (no solution). But if it's also zero, you've found identical lines!