Cramer’s rule solves a square linear system by replacing one coefficient column at a time with the constants column. This editable lesson develops the rule from elimination rather than presenting three unexplained formulas. It includes determinant foundations, fully worked two-variable and three-variable examples, singular cases, applications, 30 sequenced tasks, an exit ticket, and a separate teacher key.
The essential rule
For the system $A\mathbf{x}=\mathbf{b}$, let $D=\det(A)$. Let $D_i$ be the determinant obtained by replacing column $i$ of $A$ with $\mathbf{b}$. If $D\ne0$, the unique solution is
$$ x_i=\frac{D_i}{D}. $$
For
$$ \begin{aligned} ax+by&=e,\ cx+dy&=f, \end{aligned} $$
this becomes
$$ D=ad-bc,\qquad D_x=ed-bf,\qquad D_y=af-ec, $$
and therefore $x=D_x/D$ and $y=D_y/D$ when $D\ne0$.
Worked two-variable example
Solve $2x+3y=7$ and $x-y=1$.
$$ D=\begin{vmatrix}2&3\1&-1\end{vmatrix}=-5, \quad D_x=\begin{vmatrix}7&3\1&-1\end{vmatrix}=-10, \quad D_y=\begin{vmatrix}2&7\1&1\end{vmatrix}=-5. $$
Thus $x=2$ and $y=1$. Substitution checks both equations: $2(2)+3(1)=7$ and $2-1=1$.
What happens when the determinant is zero?
$D=0$ means only that the system does not have a unique solution. Cramer’s quotients are undefined, so use row reduction or compare equations:
- $x+2y=4$ and $2x+4y=8$ describe the same line and have infinitely many solutions.
- $x+2y=4$ and $2x+4y=10$ have proportional left sides but incompatible constants, so they have no solution.
This distinction matters: neither system may be solved by dividing by $D$.
Worked three-variable example
For
$$ \begin{aligned} x+y+z&=4,\ 2x-y+3z&=14,\ -x+2y+z&=-1, \end{aligned} $$
the coefficient determinant is $D=-9$. Replacing the first, second, and third columns gives $D_x=-18$, $D_y=9$, and $D_z=-27$. Therefore
$$ (x,y,z)=\left(\frac{-18}{-9},\frac9{-9},\frac{-27}{-9}\right)=(2,-1,3). $$
The editable project shows every cofactor expansion and verifies the result in all three original equations.
Representative tasks
- Set up $D$, $D_x$, and $D_y$ without moving the untouched coefficient column.
- Solve exact two-variable systems and verify each answer by substitution.
- Classify determinant-zero systems using equation comparison or rank.
- Analyze how the parameter $k$ changes the solution count of a system.
- Apply Cramer’s rule to homogeneous systems and explain the trivial solution.
- Solve and check three-variable systems using four determinants.
- Model price, mixture, and production contexts with linear systems.
- Diagnose sign errors, incorrect replacement columns, and invalid division by zero.
Classroom use and adaptation
Use the opening sections as direct instruction, assign selected questions for guided practice, and reserve the error-analysis tasks and exit ticket for independent practice. Teachers can shorten the packet to two-variable systems, replace contexts, change scoring and answer space, or distribute the teacher key separately.
The material targets CCSS HSA-REI.C.6, HSA-REI.C.8, and HSA-REI.C.9. It treats Cramer’s rule as a transparent exact method for small systems, not as the best numerical algorithm for large systems.
References
The lesson is adapted with permission from an original System of Linear Equations chapter by Ankit Kumar Chauhan. Standards links and mathematical references are included in the editable project. Use the lesson as a complete sequence or select only the two-variable, three-variable, or application sections that fit the class.