--- title: Cramer's Rule subtitle: Determinants, linear systems, exact solutions, and consistency author: Mathematics Department organization: Your School recipient: Algebra II or Precalculus students date: 2026-09-09 version: "1.0" description: A complete lesson and practice set on deriving, applying, checking, and limiting Cramer's rule for two-variable and three-variable systems. keywords: [Cramer's rule, systems of linear equations, determinants, matrices, singular systems, Algebra II, Precalculus] template: assignment paper: us-letter titlePage: false toc: false tocDepth: 2 headingNumbers: false equationNumbers: true columns: 1 figurePlacement: none tableCaptionPosition: top header: Cramer's Rule | Linear Systems footer: Lesson, worked examples, and practice pdfProfile: tagged language: en-US --- # Teacher launch guide > [!IMPORTANT] > **Suggested use:** two 50-minute lessons plus independent practice. Lesson 1 develops the $2\times2$ rule and singular cases. Lesson 2 extends the structure to $3\times3$ systems and applications. Remove this guide before distributing a student-only copy. ## Learning goals Students will be able to: 1. translate a square linear system into $A\mathbf{x}=\mathbf{b}$; 2. evaluate $2\times2$ and $3\times3$ determinants accurately; 3. derive Cramer's rule for two variables by elimination; 4. replace the correct coefficient column to form each numerator determinant; 5. solve and verify $2\times2$ and $3\times3$ systems when $\det(A)\ne0$; 6. explain why $\det(A)=0$ does not, by itself, distinguish no solution from infinitely many solutions; 7. choose Cramer's rule only when it is mathematically valid and practically sensible. ## Standards and evidence This lesson supports **CCSS HSA-REI.C.6, HSA-REI.C.8, and HSA-REI.C.9**. Collect determinant setups, substitution checks, singular-case explanations, and the individual exit ticket as evidence. :::table{#tbl:sequence caption="Suggested sequence and evidence of learning"} | Segment | Time | Teacher move | Evidence | | --- | ---: | --- | --- | | Read the system | 8 min | Separate coefficients from constants | Correct $A$, $\mathbf{x}$, and $\mathbf{b}$ | | Derive the rule | 15 min | Connect elimination to determinants | Explained numerator and denominator | | Work $2\times2$ cases | 20 min | Require a substitution check | Exact verified solution | | Diagnose $D=0$ | 15 min | Compare equations or row-reduce | Correct solution-count claim | | Extend to $3\times3$ | 25 min | Preserve column order | Four correct determinants | | Apply and critique | 17 min | Ask when the method is useful | Model, method choice, and interpretation | ::: **Prerequisites:** solving a two-variable system, matrix notation, signed arithmetic, and $\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$. **Likely errors:** moving the constants into the wrong column; changing the order of untouched columns; using $bc-ad$ in one determinant; dividing when $D=0$; and reporting an answer without checking the original equations. ## Prerequisite check Evaluate mentally, then compare methods with a partner. - $\begin{vmatrix}4&-3\\2&5\end{vmatrix}$ - $\begin{vmatrix}-2&7\\3&1\end{vmatrix}$ - What happens to a determinant when its two rows are interchanged? - Why is a determinant zero when one row is a multiple of another? # 1. A system is one matrix equation A **system of linear equations** asks for values that satisfy every equation at once. A square system has the same number of equations as unknowns. For two variables, :::equation{#eq:two-system alt="a x plus b y equals e, and c x plus d y equals f"} \[ \begin{aligned} ax+by&=e,\\ cx+dy&=f. \end{aligned} \] ::: Its matrix form is :::equation{#eq:matrix-form alt="the two by two coefficient matrix times the variable column equals the constants column"} \[ A=\begin{bmatrix}a&b\\c&d\end{bmatrix},\qquad \mathbf{x}=\begin{bmatrix}x\\y\end{bmatrix},\qquad \mathbf{b}=\begin{bmatrix}e\\f\end{bmatrix},\qquad A\mathbf{x}=\mathbf{b}. \] ::: The coefficient determinant is \[ D=\det(A)=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc. \] If $D\ne0$, the rows contain independent information and the system has one solution. If $D=0$, the coefficient matrix is singular: a unique solution is impossible, but the system still needs classification. > [!NOTE] > Keep $\mathbf{b}$ outside $A$. Constants replace one column only when a numerator determinant is formed. # 2. Derive the $2\times2$ rule Start from @eq:two-system. Eliminate $y$: multiply the first equation by $d$, the second by $b$, and subtract. \[ \begin{aligned} ad x+bd y&=de,\\ bc x+bd y&=bf,\\ (ad-bc)x&=de-bf. \end{aligned} \] The coefficient of $x$ is $D$. The right side is the determinant obtained by replacing the **first** coefficient column with the constants: \[ D_x=\begin{vmatrix}e&b\\f&d\end{vmatrix}=ed-bf. \] Thus $Dx=D_x$. Repeating the process to eliminate $x$ gives \[ Dy=af-ec =\begin{vmatrix}a&e\\c&f\end{vmatrix} =D_y. \] When $D\ne0$, division is valid: :::equation{#eq:cramer-2 alt="x equals D sub x over D, and y equals D sub y over D"} \[ x=\frac{D_x}{D},\qquad y=\frac{D_y}{D}. \] ::: The formula is elimination written compactly. The replaced column identifies the variable: first column for $x$, second for $y$. ## Worked example 1: exact integer solution Solve \[ 2x+3y=7,\qquad x-y=1. \] Preserve coefficient order: \[ D=\begin{vmatrix}2&3\\1&-1\end{vmatrix}=-2-3=-5. \] Replace the first column for $x$ and the second for $y$: \[ D_x=\begin{vmatrix}7&3\\1&-1\end{vmatrix}=-7-3=-10, \qquad D_y=\begin{vmatrix}2&7\\1&1\end{vmatrix}=2-7=-5. \] Therefore \[ x=\frac{-10}{-5}=2,\qquad y=\frac{-5}{-5}=1. \] **Check:** $2(2)+3(1)=7$ and $2-1=1$. Both equations hold. ## Worked example 2: fractions are exact answers Solve $3x-2y=4$ and $x+y=5$. \[ D=5,\qquad D_x=14,\qquad D_y=11. \] Hence $x=14/5$ and $y=11/5$. Do not round: the fractions are exact and substitute cleanly. # 3. What $D=0$ actually tells us When $D=0$, @eq:cramer-2 would divide by zero. Stop using the quotient and inspect the system. :::table{#tbl:singular-cases caption="Two singular systems with different outcomes"} | System | Coefficient relation | Constant relation | Conclusion | | --- | --- | --- | --- | | $x+2y=4$; $2x+4y=8$ | Row 2 is $2$ times row 1 | $8=2(4)$ | Same line; infinitely many solutions | | $x+2y=4$; $2x+4y=10$ | Row 2 is $2$ times row 1 | $10\ne2(4)$ | Parallel lines; no solution | ::: In general, compare ranks: - $\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf b])=n$: one solution; - $\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf b]) B{D nonzero?} B -- Yes --> C[Replace columns] C --> D[Divide and verify] B -- No --> E[Row-reduce augmented matrix] E --> F{Contradiction?} F -- Yes --> G[No solution] F -- No --> H[Infinitely many solutions] ``` ::: ## Parameter example Consider \[ kx+y=2,\qquad x+ky=2. \] Here $D=k^2-1=(k-1)(k+1)$. - If $k\ne\pm1$, the system has one solution: $x=y=2/(k+1)$. - If $k=1$, both equations are $x+y=2$: infinitely many solutions. - If $k=-1$, the left sides are opposites but the constants are equal: no solution. The exceptional parameter values must be checked in the original system; canceling $k-1$ too early would hide the $k=1$ case. ## Homogeneous systems A homogeneous system has $\mathbf b=\mathbf0$, so $\mathbf x=\mathbf0$ is always a solution. - If $D\ne0$, the zero vector is the only solution. - If $D=0$, at least one free variable remains, so nonzero solutions also exist. # 4. Extend the pattern to $3\times3$ For \[ A= \begin{bmatrix} a_1&b_1&c_1\\ a_2&b_2&c_2\\ a_3&b_3&c_3 \end{bmatrix}, \qquad \mathbf b= \begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix}, \] let $D=\det(A)$. Replace only the column of the variable being found: \[ D_x=\begin{vmatrix}d_1&b_1&c_1\\d_2&b_2&c_2\\d_3&b_3&c_3\end{vmatrix}, \quad D_y=\begin{vmatrix}a_1&d_1&c_1\\a_2&d_2&c_2\\a_3&d_3&c_3\end{vmatrix}, \quad D_z=\begin{vmatrix}a_1&b_1&d_1\\a_2&b_2&d_2\\a_3&b_3&d_3\end{vmatrix}. \] If $D\ne0$, :::equation{#eq:cramer-3 alt="x equals D sub x over D, y equals D sub y over D, z equals D sub z over D"} \[ x=\frac{D_x}{D},\qquad y=\frac{D_y}{D},\qquad z=\frac{D_z}{D}. \] ::: ## Why the other variables disappear Multiply the three original equations by the cofactors $C_{11}$, $C_{21}$, and $C_{31}$ of the first column, then add. The coefficient of $x$ is \[ a_1C_{11}+a_2C_{21}+a_3C_{31}=D. \] The coefficient of $y$ is $b_1C_{11}+b_2C_{21}+b_3C_{31}=0$: it is the expansion of a determinant with two equal columns. The $z$ coefficient vanishes for the same reason. The constants produce $D_x$. Thus $Dx=D_x$. The arguments for $y$ and $z$ are identical. ## Worked example 3: all four determinants Solve \[ \begin{aligned} x+y+z&=4,\\ 2x-y+3z&=14,\\ -x+2y+z&=-1. \end{aligned} \] First compute the coefficient determinant by expansion along row 1: \[ \begin{aligned} D &=\begin{vmatrix}1&1&1\\2&-1&3\\-1&2&1\end{vmatrix}\\ &=1(-1-6)-1(2+3)+1(4-1)\\ &=-7-5+3=-9. \end{aligned} \] Since $D\ne0$, one solution exists. Now replace columns: \[ \begin{aligned} D_x &=\begin{vmatrix}4&1&1\\14&-1&3\\-1&2&1\end{vmatrix} =4(-7)-(14+3)+(28-1)=-18,\\ D_y &=\begin{vmatrix}1&4&1\\2&14&3\\-1&-1&1\end{vmatrix} =(14+3)-4(2+3)+(-2+14)=9,\\ D_z &=\begin{vmatrix}1&1&4\\2&-1&14\\-1&2&-1\end{vmatrix} =(1-28)-(-2+14)+4(4-1)=-27. \end{aligned} \] Therefore \[ (x,y,z)=\left(\frac{-18}{-9},\frac9{-9},\frac{-27}{-9}\right)=(2,-1,3). \] **Check all equations:** \[ 2-1+3=4,\qquad 4+1+9=14,\qquad -2-2+3=-1. \] One correct equation is not enough; a solution must satisfy the entire system. # 5. Applications and method choice ## Worked example 4: prices Two notebooks and three pens cost USD 13. Three notebooks and two pens cost USD 12. Let $n$ and $p$ be the unit prices. \[ 2n+3p=13,\qquad 3n+2p=12. \] Then $D=-5$, $D_n=-10$, and $D_p=-15$, so $n=2$ and $p=3$. Units matter: the answer is **USD 2 per notebook and USD 3 per pen**, not merely $(2,3)$. ## Worked example 5: a mixture How many liters of 20% solution and 50% solution make 10 liters of 32% solution? Let $u$ and $v$ be liters of 20% and 50% solution: \[ u+v=10,\qquad 0.20u+0.50v=3.20. \] Cramer's rule gives $u=6$ and $v=4$. Check both constraints: $6+4=10$ and $0.20(6)+0.50(4)=3.20$ liters of solute. ## When should we use Cramer's rule? :::table{#tbl:method-choice caption="Method choice should follow the structure of the task"} | Situation | Usually useful | Reason | | --- | --- | --- | | Small exact square system | Cramer's rule | Symmetric formulas; one variable can be isolated directly | | Many systems with the same $A$ | Factorization or elimination | Repeating several determinants wastes work | | Large numerical system | Gaussian elimination or a stable solver | Cramer's rule scales poorly and can amplify rounding work | | Need to classify a singular system | Row reduction or rank | $D=0$ only rules out uniqueness | | Symbolic dependence on a parameter | Cramer's rule can help | The determinant exposes exceptional parameter values | ::: > [!TIP] > A method is not complete until its hypotheses are checked. For Cramer's rule: square system, consistent column order, $D\ne0$, and verification in the original equations. # 6. Guided and independent practice Show determinant setups. Use exact values unless a context requires a decimal. Verify every claimed unique solution. ::::regions{tracks="1 1" gutter=18} :::region ## A. Determinants and $2\times2$ systems 1. Evaluate $\begin{vmatrix}3&2\\-1&4\end{vmatrix}$. 2. For $4x-y=9$ and $2x+3y=1$, write and evaluate $D$, $D_x$, and $D_y$. Then solve. 3. Solve $x+2y=8$ and $3x-y=3$ by Cramer's rule. 4. Solve $5x+2y=1$ and $-x+3y=10$. 5. Explain why the untouched coefficient column must stay in its original position when $D_x$ or $D_y$ is formed. ::: :::region ## B. Singular and homogeneous systems 6. Classify $x-3y=2$ and $2x-6y=4$. Give a parametric solution if appropriate. 7. Classify $x-3y=2$ and $2x-6y=5$. 8. For $kx+y=2$ and $x+ky=2$, classify the solution count for every real $k$. 9. Solve the homogeneous system $3x+2y=0$ and $x-y=0$. 10. Describe all solutions of $x+2y=0$ and $2x+4y=0$. ::: :::: ## C. Three-variable systems 11. Evaluate $\begin{vmatrix}1&2&0\\3&-1&2\\2&1&1\end{vmatrix}$. 12. Use Cramer's rule to solve \[ \begin{aligned} x+y+z&=6,\\ 2x-y+z&=3,\\ x+2y-z&=2. \end{aligned} \] 13. Use Cramer's rule to solve \[ \begin{aligned} 2x+y-z&=1,\\ x-2y+3z&=10,\\ 3x+y+2z&=9. \end{aligned} \] 14. Classify and describe the solution set: \[ x+y+z=3,\quad 2x+2y+2z=6,\quad x-y+z=1. \] 15. Classify the system obtained from question 14 by changing the second constant from 6 to 7. 16. Find all solutions of the homogeneous system \[ x+y=0,\quad 2x-y+z=0,\quad 3x+z=0. \] ## D. Model and interpret 17. Two notebooks and three pens cost USD 13; three notebooks and two pens cost USD 12. Find both unit prices and state the units. 18. Determine the liters of 20% and 50% solutions needed for 10 liters of a 32% solution. State both physical constraints used in the model. 19. A theater sold 180 tickets. Adult tickets cost USD 12 and student tickets cost USD 7; revenue was USD 1,660. How many of each were sold? 20. A workshop makes 40 objects, some type A and the rest type B. Type A uses 3 machine-hours and type B uses 2; total machine time is 95 hours. Find the counts and check that they are nonnegative integers. 21. Write a two-variable context whose coefficient determinant is zero. Give one version with infinitely many solutions and one with no solution. Explain what each outcome means in context. ## E. Analyze errors and structure 22. Jordan forms $D_x$ by replacing both columns of $A$ with the constants column. Explain why the determinant becomes zero and correct the procedure. 23. A student solves $3x-2y=4$, $x+y=5$ and writes $D=3(1)-2(1)=1$. Identify the sign error and finish the solution. 24. Refute the claim: "If $D=0$, the system has no solution." Give two explicit counterexamples with different outcomes. 25. For a $3\times3$ system with small integer coefficients, when might elimination be preferable to Cramer's rule? Name the work each method requires. 26. Analyze $px+y=1$ and $x+py=1$ for every real $p$. Do not cancel a factor until exceptional values have been checked. 27. Suppose $A\mathbf{x}=\mathbf b$ has $D\ne0$. Explain, using replacement determinants, why replacing $\mathbf b$ by $5\mathbf b$ multiplies every solution coordinate by 5. 28. Create a reliable one-line mnemonic for forming $D_x$, $D_y$, and $D_z$. Test it on a system with variables written in a nonstandard order. 29. A classmate says, "Cramer's rule is always faster because it is a formula." Give a mathematically specific response for a system with many variables. 30. The determinant of a $2\times2$ coefficient matrix is very close to zero but not zero. Explain why a unique solution exists and why small data changes may still cause large changes in that solution. # 7. Exit ticket Complete without notes. 1. State the hypothesis that permits division in Cramer's rule. 2. Form $D_y$ for $2x+5y=9$ and $-x+4y=3$. 3. Solve $x+y=7$ and $2x-y=2$. 4. What can be concluded immediately when $D=0$? What cannot be concluded? 5. For a homogeneous square system with $D\ne0$, state the complete solution set. 6. Name one reason to prefer row reduction over Cramer's rule. ## Student reflection - The step I can now justify, rather than memorize, is: ____________________ - The error I am most likely to catch during verification is: ______________ - One question I still have about singular systems is: _____________________ # Sources for teacher review - [Common Core State Standards: Reasoning with Equations and Inequalities](https://www.thecorestandards.org/Math/Content/HSA/REI/) - [OpenStax, *Precalculus 2e*](https://openstax.org/details/books/precalculus-2e) - Original *System of Linear Equations* chapter by Ankit Kumar Chauhan, adapted with the author's permission. > [!NOTE] > This publication example has been checked for mathematical consistency and page layout. Before classroom distribution, select tasks and adjust timing, scoring, and answer space for your students and local curriculum.