Expresses each unknown of a square system as a ratio of determinants, with the coefficient determinant as the common denominator and a column replaced by the constants in each numerator. Extends to three variables, and reads a vanishing denominator as the singular case.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 9 — Systems of Equations and Inequalities
§9.8 Solving Systems with Cramer's Rule, pp. 1161-1174
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1161-1174 — the pages these objectives are drawn from
Warm-up
Elimination finds them all, whether you need them or not.
Discussion prompt
A system has three unknowns and you need only the second. What does row reduction make you do?
Hint: How much of the work is wasted?
Answer:
Row reduction drives the whole matrix to echelon form and then back-substitutes, so it produces all three values on the way to the one you wanted.
Most of that work is wasted if only one unknown matters — and back-substitution has to run through the others to reach it.
Cramer's rule gives each unknown by a formula of its own, so one can be computed without touching the rest. That is its whole appeal.
Concept
For a square system with a nonzero coefficient determinant, each unknown equals a determinant with its column replaced by the constants, divided by the coefficient determinant.
\[ x=\frac{D_x}{D}, \quad y=\frac{D_y}{D} \]
The denominator is the same for every unknown, so it is computed once. Only the numerator changes, and it changes by which column gets replaced — which is what ties each formula to its own variable.
Figure (svg): A diagram showing the coefficient determinant and the determinant formed by replacing one column with the constants
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1161-1165
Section
Section 1
Concept
The denominator is the determinant of the coefficient matrix. Each numerator is that determinant with the column belonging to the unknown replaced by the constants.
Replacing the variable's own column is the point to get right. Replacing the other one computes the wrong unknown, and since both ratios are plausible-looking numbers, nothing about the answer signals the swap.
Figure (svg): A diagram showing the coefficient determinant and the determinant formed by replacing one column with the constants
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1161-1166
Picture it
The same matrix with one column swapped each time.
Figure (svg): A diagram showing the coefficient determinant and the determinant formed by replacing one column with the constants
The shaded column in each numerator is the one holding the constants. Which column that is determines which unknown the ratio gives.
Worked example
Three determinants, two divisions.
\[ \text{Solve } 2x+5y=11, \; x+3y=6 \text{ by Cramer's rule.} \]
Compute the denominator
Why: Coefficient determinant.
\[ 6 - 5 = 1 \]
Form the first numerator
Why: Constants in column one.
\[ 33 - 30 = 3 \]
Form the second numerator
Why: Constants in column two.
\[ 12 - 11 = 1 \]
Divide
Why: Each numerator by the denominator.
\[ x = 3, y = 1 \]
Figure (svg): A diagram showing the coefficient determinant and the determinant formed by replacing one column with the constants
\[ (x,y)=(3,1) \]
Verify: substitute into both equations
Why: Six plus five is 11 — correct. Three plus three is 6 — also correct. Three determinants replaced the whole reduction, and either unknown could have been computed on its own.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1162-1164
Prediction
You want the second unknown.
Predict first
Which column gets the constants?
Correct: The second.
Why: Each unknown corresponds to its own column of coefficients, and that is the column replaced in its numerator. Replacing a different one computes a different unknown while appearing to answer the question asked.
Worked example
The other never has to be found.
\[ \text{Find only } y \text{ for } 3x+4y=18, \; 5x-2y=4. \]
Compute the denominator
Why: Coefficient determinant.
\[ -6 - 20 = -26 \]
Replace the second column
Why: The one belonging to the second unknown.
\[ 12 - 90 = -78 \]
Divide
Why: Numerator over denominator.
\[ -78 / - 26 \]
State the answer
Why: One value.
\[ y = 3 \]
Figure (svg): The solution to Worked example one unknown only shown as a ladder of expressions, one row per legal move
\[ y=3 \]
Verify: check by substituting into one equation
Why: In the first equation, 3x plus 12 equals 18 gives x equal to 2, and checking that pair in the second gives 10 minus 6, which is 4 — correct. The check needed the other unknown but the computation did not, which is exactly the saving Cramer's rule offers.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1164-1166
Trap
\[ y=\frac{\text{first column replaced}}{D} \]
Replace whichever column is convenient
Why: The correspondence between column and variable is not checked.
The ratio computes the first unknown while being labelled as the second.
Each unknown's column is the one that gets replaced. The second unknown needs the second column.
The columns correspond to variables in the same order they appear in the equations, exactly as in §9.6's augmented matrix.
Nothing about the answer reveals the swap, since both ratios are ordinary numbers. Checking which column belongs to which variable before computing is the only defence.
Faded example
Replacing the first column with the constants 11 and 6.
Fill in the blanks
D_x=11(3)-5(6)=33-30=3
Why: The constants take the place of the first column and the second column stays as it was. The determinant is then computed exactly as usual, as a difference of diagonal products.
Sorting
One is shared and two are specific.
Sort into buckets
Sort each determinant.
Step zero
You are given a system to solve by Cramer's rule.
Discussion prompt
What do you compute first?
Hint: One number decides whether the method works.
Answer:
The coefficient determinant, because it is the denominator for every unknown and because a zero value means the rule cannot be used at all.
Computing it first means a singular system is identified before any numerator work is done.
And it is reused for every unknown, so it is the one determinant always worth having. Computing a numerator first risks doing work that a zero denominator will waste.
Section
Section 2
Concept
A three-by-three determinant is computed by taking each entry of one row, multiplying by the determinant of what remains after deleting its row and column, and alternating the signs.
Any row or column may be used and all give the same value, which is worth knowing because choosing one containing zeros can eliminate whole terms. A row with two zeros reduces the work to a single minor.
Figure (svg): A card showing the expansion of a three-by-three determinant along its first row into three two-by-two minors
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1166-1170
Picture it
Three two-by-two determinants, with alternating signs.
Figure (svg): A card showing the expansion of a three-by-three determinant along its first row into three two-by-two minors
The red line at the bottom flags the commonest slip. The middle term is subtracted, and dropping that sign is the single most frequent error in the expansion.
Worked example
Three minors, alternating signs.
\[ \text{Evaluate the determinant of } \begin{bmatrix}1&2&3\\0&1&4\\5&6&0\end{bmatrix}. \]
First entry and its minor
Why: Delete row 1, column 1.
\[ 1 \times(0 - 24) \]
Second entry, subtracted
Why: Delete row 1, column 2.
\[ \text{minus } 2 \times(0 - 20) \]
Third entry, added
Why: Delete row 1, column 3.
\[ \text{plus } 3 \times(0 - 5) \]
Combine
Why: Sum the three terms.
\[ -24 + 40 - 15 \]
Figure (svg): A card showing the expansion of a three-by-three determinant along its first row into three two-by-two minors
\[ 1 \]
Verify: expand along a different row
Why: Expanding along the second row, which contains a zero, gives negative one times the minor of the middle entry plus four times the last — and the same total of 1. Every row and column gives the same value, so this is a genuine independent check.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1167-1169
Prediction
Expanding along the first row of a three-by-three determinant.
Predict first
What signs do the three terms take?
Correct: Plus, minus, plus.
Why: The signs alternate in a checkerboard beginning with a plus at the top left. Dropping the middle minus sign is the commonest error and changes the value by twice that term.
Worked example
A zero kills a whole term.
\[ \text{Which row should you expand } \begin{bmatrix}2&1&7\\0&0&3\\4&5&6\end{bmatrix} \text{ along?} \]
Scan for zeros
Why: The second row has two.
\[ r o w 2 \]
Note what they eliminate
Why: Two terms vanish.
Expand along it
Why: Only the third entry contributes.
\[ 3 \times\text{ its minor} \]
Compute
Why: With the sign for that position.
\[ -3(10 - 4) = -18 \]
Figure (svg): The solution to Worked example choose a row with zeros shown as a ladder of expressions, one row per legal move
\[ -18 \]
Verify: confirm the sign for that position
Why: The entry sits in the second row and third column, and the sign pattern alternates in a checkerboard from a plus at the top left — so that position takes a minus. Expanding along the first row instead would give the same answer with three times the arithmetic.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1169-1170
Error analysis
A student expands a three-by-three determinant.
Annotate
On: \( a(ei-fh)+b(di-fg)+c(dh-eg) \)
The alternating signs are part of the definition rather than a convention that can be absorbed into the minors. Expanding along a different row and comparing is the check, since every row must give the same value.
Faded example
Deleting the first row and first column.
Fill in the blanks
\text2a\text2___\times___
Why: Deleting one row and one column from a three-by-three matrix leaves a two-by-two one, whose determinant is computed as a difference of diagonal products. Each entry of the expansion row has its own minor.
Sorting
Zeros eliminate terms.
Sort into buckets
Sort each row by how much work it saves.
Explain it to yourself
The expansion can be done along any row or column.
Discussion prompt
Explain what that means for checking your work.
Hint: What should two expansions agree on?
Answer:
Every row and every column gives the same value, which is a theorem rather than a coincidence. So the choice of expansion line is purely about convenience.
That makes a second expansion a genuine independent check: computing along a different row uses different minors and different arithmetic, so agreement is meaningful.
And it gives a strategy: choose the line with the most zeros, since each zero eliminates a whole minor. A good explanation notes that this can cut the work to a third of the general case.
Section
Section 3
Concept
The pattern extends unchanged: one coefficient determinant as the shared denominator, and one numerator per unknown formed by replacing that unknown's column.
Four three-by-three determinants is a lot of arithmetic, which is why the rule is rarely the fastest route to all three unknowns. Its advantage survives only when one unknown is wanted, where it needs two determinants rather than four.
Figure (svg): A contrast between the situations where Cramer's rule is the better method and those where row reduction is
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1170-1172
Picture it
The left column is Cramer's territory.
Figure (svg): A contrast between the situations where Cramer's rule is the better method and those where row reduction is
The first row on each side is the decisive one. Wanting one unknown favours the formula and wanting all of them favours the algorithm.
Worked example
Two determinants rather than four.
\[ \text{Find } z \text{ for } x+y+z=6, \; 2x-y+z=3, \; x+2y-z=2. \]
Compute the denominator
Why: The coefficient determinant.
\[ 7 \]
Replace the third column
Why: The one belonging to the third unknown.
\[ \text{constants in column } 3 \]
Expand that determinant
Why: By minors.
\[ 21 \]
Divide
Why: Numerator over denominator.
\[ z = 3 \]
Figure (svg): The solution to Worked example one unknown of three shown as a ladder of expressions, one row per legal move
\[ z=3 \]
Verify: substitute into one equation
Why: The system's solution is known from §9.2 to be one, two and three, so the third unknown being 3 is correct. Two determinants gave it directly, where row reduction would have produced all three values on the way.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1170-1171
Prediction
A three-variable system, all unknowns wanted.
Predict first
How many determinants does Cramer's rule need?
Correct: Four.
Why: One coefficient determinant serves as the shared denominator, and each of the three unknowns needs its own numerator. Wanting only one unknown would need just two.
Worked example
Compare the two methods honestly.
\[ \text{Compare the work for all three unknowns by each method.} \]
Count Cramer's determinants
Why: One denominator, three numerators.
\[ \text{four } 3 x 3\text{ determinants} \]
Count the minors
Why: Three per determinant.
\[ \text{twelve } 2 x 2\text{ determinants} \]
Count row reduction's operations
Why: A handful of row operations.
Conclude
Why: For all three, reduction wins.
Figure (svg): A contrast between the situations where Cramer's rule is the better method and those where row reduction is
\[ \text{reduction for all; Cramer for one} \]
Verify: check where the crossover is
Why: For a single unknown, Cramer's needs two determinants — six two-by-two minors — where row reduction still needs the full reduction and a back-substitution. So the advantage genuinely reverses depending on how many unknowns are wanted.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1171-1172
Trap
\[ \text{a four-variable system, so compute five }4\times 4\text{ determinants} \]
Apply the rule because it is a formula
Why: The arithmetic cost is not compared against the alternative.
Each four-by-four determinant needs four three-by-three expansions, and the work explodes.
The work grows very fast with size. Each extra variable multiplies the determinant cost several times over.
Row reduction's cost grows far more slowly and handles the singular cases as well.
Cramer's rule is for small systems or single unknowns. Beyond three variables it is a theoretical tool rather than a practical one.
Sorting
How many unknowns are wanted decides.
Sort into buckets
Sort each situation.
Faded example
For one unknown of a three-variable system.
Fill in the blanks
1\text1+___\text___=2
Why: Only the wanted unknown's numerator has to be formed, alongside the shared denominator. That is half the work of finding two unknowns and a quarter of finding all three.
Explain it
Both methods solve the same systems.
Discussion prompt
Explain to a classmate when each is the better choice.
Hint: How many answers do you need?
Answer:
Cramer's rule gives each unknown by its own formula, so one can be computed without the others. That is the case where it wins.
Row reduction produces all the unknowns as a single process, so when all are wanted its work is shared rather than repeated.
And the sizes matter: Cramer's cost grows very fast with the number of variables, since each determinant expands into several smaller ones. A good explanation notes that this is why it is standard for two and three variables and essentially never used beyond that.
Section
Section 4
Concept
When the coefficient determinant is zero, every ratio is a division by zero and the rule produces nothing. That is exactly the singular case of the previous section.
The last point is worth noting: a zero over zero is indeterminate rather than impossible, and it corresponds to the dependent case. A nonzero numerator over a zero denominator is a genuine impossibility and signals inconsistency.
Figure (svg): A card showing that a zero coefficient determinant makes every Cramer ratio undefined, and what that signals about the system
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1172-1174
Picture it
Every ratio becomes undefined at once.
Figure (svg): A card showing that a zero coefficient determinant makes every Cramer ratio undefined, and what that signals about the system
The bottom line connects this to §9.7. The same singular matrix that had no inverse produces the zero denominator here, because both are the determinant vanishing.
Worked example
Zero denominator, nonzero numerator.
\[ \text{Apply Cramer's rule to } 3x+6y=9, \; x+2y=4. \]
Compute the denominator
Why: Coefficient determinant.
\[ 6 - 6 = 0 \]
Note the division fails
Why: Every ratio is undefined.
Compute a numerator
Why: Replace the first column.
\[ 18 - 24 = -6 \]
Interpret
Why: Nonzero over zero.
Figure (svg): A card showing that a zero coefficient determinant makes every Cramer ratio undefined, and what that signals about the system
\[ \text{inconsistent: no solution} \]
Verify: check against the equations
Why: Three times the second equation is 3x plus 6y equals 12, which contradicts the first equation's 9. The lines are parallel and distinct, which is what a nonzero numerator over a zero denominator signals.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1173-1174
Prediction
The coefficient determinant is zero.
Predict first
What does Cramer's rule give?
Correct: Nothing; every ratio is undefined.
Why: The denominator is shared by every unknown, so all the ratios fail at once. The system has none or infinitely many solutions, and the numerators decide which.
Worked example
Both zero this time.
\[ \text{Apply Cramer's rule to } 3x+6y=12, \; x+2y=4. \]
Compute the denominator
Why: Same coefficients.
\[ 0 \]
Compute the first numerator
Why: Replace column one.
\[ 24 - 24 = 0 \]
Compute the second numerator
Why: Replace column two.
\[ 12 - 12 = 0 \]
Interpret
Why: Zero over zero throughout.
Figure (svg): The solution to Worked example a dependent system shown as a ladder of expressions, one row per legal move
\[ \text{dependent: infinitely many} \]
Verify: compare with the previous example
Why: The two systems have identical coefficient matrices and identical zero denominators, but different numerators — nonzero in one case and zero in the other. That difference is what distinguishes the two singular outcomes, and it comes entirely from the constants.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1174-1174
Error analysis
A student computes a ratio with a zero denominator.
Annotate
On: \( x=\frac{-6}{0}=0 \)
Confusing a zero numerator with a zero denominator inverts the meaning entirely. A zero numerator gives an unknown equal to zero; a zero denominator gives no unknown at all.
Sorting
Numerator and denominator play different roles.
Sort into buckets
Sort each situation.
Faded example
The denominator and every numerator come out zero.
Fill in the blanks
D=0\text0D_i=___ \;\Longrightarrow\; \text___
Why: All zeros throughout corresponds to the dependent case, where the equations describe the same line. A nonzero numerator with a zero denominator would signal inconsistency instead.
Explain it
The same determinant appeared there.
Discussion prompt
Explain to a classmate what links the two sections.
Hint: What does the determinant measure?
Answer:
The same number appears in both: the coefficient determinant. In §9.7 it was the divisor in the inverse formula and here it is the denominator of every ratio.
So a zero value breaks both methods for the same reason — the coefficient rows are dependent, and no unique solution exists to be found.
It is one fact wearing two costumes. A good explanation notes that this is why the determinant is worth computing first in either method: it settles whether a unique solution exists before any other work is done.
Section
Section 5
Concept
Substitution, elimination, row reduction, inverses and Cramer's rule all solve the same systems. Which is quickest depends on the size, on how many unknowns are wanted, and on whether the coefficient matrix is reused.
All five give the same answer, so the choice is about effort rather than correctness. Recognising which situation you are in before starting is worth more than fluency in any one method.
| situation | best method |
|---|---|
| two variables, something isolated | substitution |
| two or three variables, all wanted | elimination or row reduction |
| one unknown of a small system | Cramer's rule |
| many systems, same coefficients | the inverse, computed once |
| a large system | row reduction |
Figure (svg): A contrast between the situations where Cramer's rule is the better method and those where row reduction is
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1161-1174
Picture it
The two philosophies of the chapter's later sections.
Figure (svg): A contrast between the situations where Cramer's rule is the better method and those where row reduction is
The last row on each side is the honest summary: the formula fails outright on singular systems, and the algorithm handles them and tells you which case you are in.
Worked example
Read the situation first.
\[ \text{You need only } y \text{ from a three-variable system. Which method?} \]
Note how many unknowns are wanted
Why: One of three.
Note the size
Why: Small enough for determinants.
\[ 3\text{ by } 3 \]
Compare the work
Why: Two determinants against a full reduction.
Choose
Why: The formula.
Figure (svg): A contrast between the situations where Cramer's rule is the better method and those where row reduction is
\[ \text{Cramer's rule} \]
Verify: check the alternative's cost
Why: Row reduction would drive the whole matrix to echelon form and back-substitute through the other two unknowns to reach this one — all of that work discarded. The formula's advantage is real here and disappears if all three are wanted.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1165-1172
Matching
Each has one circumstance where it wins.
Match the pairs
Why: Every method solves every non-singular system, so the choice is about effort. Recognising the situation before starting saves more time than fluency in any single method.
Worked example
A singular system.
\[ \text{The determinant is zero. Which method should you use?} \]
Rule out Cramer's rule
Why: Every ratio is undefined.
Rule out the inverse
Why: No inverse exists.
Consider row reduction
Why: It reaches echelon form regardless.
Note what it gives
Why: The special row identifies the case.
Figure (svg): The solution to Worked example when no method has an advantage shown as a ladder of expressions, one row per legal move
\[ \text{row reduction} \]
Verify: note what the other methods leave out
Why: Both Cramer's rule and the inverse method simply stop, telling you a unique solution does not exist but not whether there are none or infinitely many. Row reduction produces a special row that answers that too, which makes it the only complete method.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1172-1174
Trap
\[ \text{Cramer's rule always works, so use it for everything} \]
Apply the most recently learned method by default
Why: The situation is not considered.
Large systems become impractical and singular ones produce no answer at all.
Read the situation first. How many unknowns are wanted, how large the system is, and whether the coefficients will be reused.
Cramer's rule is excellent for one unknown of a small system and poor for everything else.
Row reduction is the safe default, since it works at any size and handles the singular cases the other methods cannot.
Prediction
You do not know whether the system is singular.
Predict first
Which method should you reach for?
Correct: Row reduction, which handles every case.
Why: Both Cramer's rule and the inverse method stop producing answers when the determinant vanishes. Row reduction reaches echelon form regardless and its special rows identify which singular case applies.
Sorting
Some stop and some diagnose.
Sort into buckets
Sort each method.
Explain it to yourself
They all solve the same systems.
Discussion prompt
Explain what having several is for.
Hint: What differs between them?
Answer:
They differ in effort, not in correctness. Each is quickest in a particular situation and slower in the others.
Cramer's rule isolates one unknown; the inverse is reused across many right-hand sides; row reduction scales and diagnoses; substitution is fastest when something is already isolated.
So the skill is recognising the situation, not mastering a single technique. A good explanation notes that this pattern recurs throughout mathematics — several tools for one job, chosen by circumstance rather than by preference.
Comparison
Fill the blanks from memory. Both solve the same systems.
Comparison matrix
| Cramer's rule | row reduction | |
|---|---|---|
| kind of thing | a formula | an algorithm |
| one unknown alone | two determinants | the full reduction anyway |
| all unknowns | one determinant per unknown, plus one | one reduction |
| singular systems | no answer at all | reaches echelon form and diagnoses |
The last row is why row reduction is the safe default. The formula's failure is silent about which singular case applies, where the algorithm's special rows say directly.
Pattern
Five steps, and the second decides whether the rest is possible.
Step 2 comes before any numerator work, so a singular system is identified before effort is wasted. Step 3's phrase 'its own column' is the one to get right.
OpenStax Algebra and Trigonometry 2e, §11.8 Solving Systems with Cramer's Rule §11.8
Check
Forming the numerators.
Check your understanding
To find the second unknown, which column is replaced by the constants?
Answer: A
Why: Each unknown corresponds to its own column of coefficients, and that column is the one replaced in its numerator. Replacing another computes a different unknown while appearing to answer the question asked.
Check
Three-by-three determinants.
Check your understanding
When expanding along the first row, what signs do the three terms take?
Answer: A
Why: The signs alternate in a checkerboard beginning with a plus at the top left. Dropping the middle minus is the commonest error in the expansion and changes the value by twice that term.
Check
The singular case.
Check your understanding
The coefficient determinant is zero. What does Cramer's rule give?
Answer: A
Why: The zero denominator makes every ratio undefined, so the rule produces nothing. The system has none or infinitely many solutions, and only the numerators — or a row reduction — decide which.
Real world
Circuit analysis often needs one current out of many.
Discussion prompt
An engineer analysing a circuit wants the current in a single branch. Why might Cramer's rule suit?
Hint: How many of the unknowns are needed?
Answer:
The circuit's loop equations form a linear system with one unknown current per loop, and the engineer may need only one of them — the current through a particular component.
Cramer's rule gives that current directly as a ratio of two determinants, without computing any of the others.
It also makes the dependence visible: the current is a formula in the resistances and voltages, so how it responds to changing one component can be read off rather than recomputed. A numerical reduction gives the number and not the relationship, which is why symbolic work in circuit theory still uses determinants.
Commit first
State your confidence along with your answer.
Predict first
What is Cramer's rule's main advantage?
Correct: It gives one unknown without computing the others.
Why: Each unknown has its own formula, so a single value can be found with two determinants. Its disadvantages are the opposite of the other options: the work grows fast with size, and it produces nothing at all when the determinant vanishes.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
A classmate got a zero denominator and reported the answer as zero. Explain the error.
Hint: Which part being zero gives zero?
Answer:
Dividing by zero is undefined, not zero. The rule produces no value at all in that case.
Zero would be the answer if the numerator were zero and the denominator were not — the two are opposite situations with opposite meanings.
And the zero denominator is informative: it says the system is singular, with either no solution or infinitely many. A good explanation adds that the numerators then decide which, or that a row reduction settles it directly.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The second is where most of the arithmetic errors happen, particularly the middle sign. The first contains the one conceptual point — that each unknown owns a column.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Draw the three two-by-two determinants of Cramer's rule, shading the replaced column in each and writing which unknown it gives. Beside them, expand one three-by-three determinant by minors with the signs marked. Underneath, write what a zero denominator means and complete a small table of which method to use when.
If your shading shows each unknown owning its own column, and your table explains why row reduction is the safe default, the chapter's methods are organised rather than merely collected.
Recap
Five things, and the last places this method among the others.
| if you remember one thing | it should be this |
|---|---|
| about the numerators | each unknown's own column is the one replaced |
| about expansion | the signs alternate, and the middle minus is the one that goes missing |
| about the denominator | zero means no answer, not an answer of zero |
| about choosing | Cramer for one unknown, row reduction for everything else |
Chapter 10 turns to the conic sections — ellipses, hyperbolas and parabolas — where the quadratic equations of chapter 3 acquire a geometric meaning as slices of a cone.
OpenStax, Precalculus, §9.8 Solving Systems with Cramer's Rule §9.8, pp. 1161-1174 — everything on these slides traces back here
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