9.9: Solving Systems with Cramer's Rule (2024)

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Learning Objectives

In this section, you will:

Evaluate 2 × 2 determinants.
Use Cramer’s Rule to solve a system of equations in twovariables.
Evaluate 3 × 3 determinants.
Use Cramer’s Rule to solve a system of three equations in threevariables.
Know the properties of determinants.

We have learned how to solve systems of equations in twovariables and three variables, and by multiple methods:substitution, addition, Gaussian elimination, using the inverse ofa matrix, and graphing. Some of these methods are easier to applythan others and are more appropriate in certain situations. In thissection, we will study two more strategies for solving systems ofequations.

Evaluating the Determinant of a 2×2Matrix

A determinant is a real number that can be very useful inmathematics because it has multiple applications, such ascalculating area, volume, and other quantities. Here, we will usedeterminants to reveal whether a matrix is invertible by using theentries of a square matrix to determine whether there isa solution to the system of equations. Perhaps one of the moreinteresting applications, however, is their use in cryptography.Secure signals or messages are sometimes sent encoded in a matrix.The data can only be decrypted with an invertiblematrix and the determinant. For our purposes, we focus on thedeterminant as an indication of the invertibility of the matrix.Calculating the determinant of a matrix involves following thespecific patterns that are outlined in this section.

FINDTHE DETERMINANT OF A 2 × 2 MATRIX

The determinant ofa 2×22×2 matrix, given

A=[acbd]A=[ abcd ]

is defined as

Notice the change in notation. There are several ways toindicate the determinant, including det(A)det(A) andreplacing the brackets in a matrix with straightlines, |A|.| A |.

EXAMPLE 1

Finding the Determinant of a 2 × 2Matrix

Find the determinant of the given matrix.

A=[5−623]A=[ 52−63 ]

Answer

Using Cramer’s Rule to Solve a Systemof Two Equations in Two Variables

We will now introduce a final method for solving systems ofequations that uses determinants. Known as Cramer’s Rule, thistechnique dates back to the middle of the 18th century and is namedfor its innovator, the Swiss mathematician Gabriel Cramer(1704-1752), who introduced it in 1750 in Introduction àl'Analyse des lignes Courbes algébriques. Cramer’s Rule is a viableand efficient method for finding solutions to systems with anarbitrary number of unknowns, provided that we have the same numberof equations as unknowns.

Cramer’s Rule will give us the unique solution to a system ofequations, if it exists. However, if the system has no solution oran infinite number of solutions, this will be indicated by adeterminant of zero. To find out if the system is inconsistent ordependent, another method, such as elimination, will have to beused.

To understand Cramer’s Rule, let’s look closely at how we solvesystems of linear equations using basic row operations. Consider asystem of two equations in two variables.

a1x+b1y=c1(1)a2x+b2y=c2(2)a1x+b1y=c1(1)a2x+b2y=c2(2)

We eliminate one variable using row operations and solve for theother. Say that we wish to solve for x.x. If equation (2)is multiplied by the opposite of the coefficient of yy inequation (1), equation (1) is multiplied by the coefficientof yy in equation (2), and we add the two equations, thevariable yy will be eliminated.

b2a1x+b2b1y=b2c1−b1a2x−b1b2y=−b1c2Multiply R1by b2Multiply R2by−b1________________________________________________________ b2a1x−b1a2x=b2c1−b1c2b2a1x+b2b1y=b2c1Multiply R1by b2−b1a2x−b1b2y=−b1c2Multiply R2by−b1________________________________________________________ b2a1x−b1a2x=b2c1−b1c2

Now, solve for x.x.

b2a1x−b1a2x=b2c1−b1c2x(b2a1−b1a2)=b2c1−b1c2 x=b2c1−b1c2b2a1−b1a2=∣∣∣c1c2b1b2∣∣∣∣∣∣a1a2b1b2∣∣∣b2a1x−b1a2x=b2c1−b1c2x(b2a1−b1a2)=b2c1−b1c2 x=b2c1−b1c2b2a1−b1a2=| c1b1c2b2 || a1b1a2b2 |

Similarly, to solve for y,y, we willeliminate x.x.

a2a1x+a2b1y=a2c1−a1a2x−a1b2y=−a1c2Multiply R1by a2Multiply R2by−a1________________________________________________________a2b1y−a1b2y=a2c1−a1c2a2a1x+a2b1y=a2c1Multiply R1by a2−a1a2x−a1b2y=−a1c2Multiply R2by−a1________________________________________________________a2b1y−a1b2y=a2c1−a1c2

Solving for yy gives

a2b1y−a1b2y=a2c1−a1c2y(a2b1−a1b2)=a2c1−a1c2 y=a2c1−a1c2a2b1−a1b2=a1c2−a2c1a1b2−a2b1=∣∣∣a1a2c1c2∣∣∣∣∣∣a1a2b1b2∣∣∣a2b1y−a1b2y=a2c1−a1c2y(a2b1−a1b2)=a2c1−a1c2 y=a2c1−a1c2a2b1−a1b2=a1c2−a2c1a1b2−a2b1=|a1c1a2c2||a1b1a2b2|

Notice that the denominator forboth xx and yy is the determinant of thecoefficient matrix.

We can use these formulas to solvefor xx and y,y, but Cramer’s Rule alsointroduces new notation:

D:D: determinant of the coefficient matrix
Dx:Dx: determinant of the numerator in the solutionof xx
x=DxDx=DxD
Dy:Dy: determinant of the numerator in the solutionof yy
y=DyDy=DyD

The key to Cramer’s Rule is replacing the variable column ofinterest with the constant column and calculating the determinants.We can then express xx and yy as a quotient oftwo determinants.

CRAMER’S RULE FOR 2×2SYSTEMS

Cramer’s Rule is a method that usesdeterminants to solve systems of equations that have the samenumber of equations as variables.

Consider a system of two linear equations in two variables.

a1x+b1y=c1a2x+b2y=c2a1x+b1y=c1a2x+b2y=c2

The solution using Cramer’s Rule is given as

x=DxD=∣∣∣c1c2b1b2∣∣∣∣∣∣a1a2b1b2∣∣∣,D≠0;y=DyD=∣∣∣a1a2c1c2∣∣∣∣∣∣a1a2b1b2∣∣∣,D≠0.x=DxD=| c1b1c2b2 || a1b1a2b2 |,D≠0;y=DyD=| a1c1a2c2 || a1b1a2b2 |,D≠0.

If we are solving for x,x, the xx column isreplaced with the constant column. If we are solvingfor y,y, the yy column is replaced with theconstant column.

EXAMPLE 2

Using Cramer’s Rule to Solve a 2 × 2System

Solve the following 2×22×2 system using Cramer’sRule.

12x+3y=15 2x−3y=1312x+3y=15 2x−3y=13

Answer

TRYIT #1

Use Cramer’s Rule to solve the 2 × 2 system of equations.

x+2y=−11−2x+y=−13 x+2y=−11−2x+y=−13

Evaluating the Determinant of a 3 × 3Matrix

Finding the determinant of a 2×2 matrix is straightforward, butfinding the determinant of a 3×3 matrix is more complicated. Onemethod is to augment the 3×3 matrix with a repetition of the firsttwo columns, giving a 3×5 matrix. Then we calculate the sum of theproducts of entries down each of the three diagonals (upper left tolower right), and subtract the products ofentries up each of thethree diagonals (lower left to upper right). This is more easilyunderstood with a visual and an example.

Find the determinant of the 3×3 matrix.

A=⎡⎣⎢a1a2a3b1b2b3c1c2c3⎤⎦⎥A=[ a1b1c1a2b2c2a3b3c3 ]

Augment AA with the first two columns.
det(A)=∣∣∣∣a1a2a3b1b2b3c1c2c3∣∣∣∣a1a2a3b1b2b3∣∣∣∣det(A)=| a1b1c1a2b2c2a3b3c3|a1a2a3b1b2b3 |
From upper left to lower right: Multiply the entries down thefirst diagonal. Add the result to the product of entries down thesecond diagonal. Add this result to the product of the entries downthe third diagonal.
From lower left to upper right: Subtract the product of entriesup the first diagonal. From this result subtract the product ofentries up the second diagonal. From this result, subtract theproduct of entries up the third diagonal.

The algebra is as follows:

|A|=a1b2c3+b1c2a3+c1a2b3−a3b2c1−b3c2a1−c3a2b1| A |=a1b2c3+b1c2a3+c1a2b3−a3b2c1−b3c2a1−c3a2b1

EXAMPLE 3

Finding the Determinant of a 3 × 3Matrix

Find the determinant of the 3 × 3 matrix given

A=⎡⎣⎢0342−10111⎤⎦⎥A=[ 0213−11401 ]

Answer

TRYIT #2

Find the determinant of the 3 × 3 matrix.

det(A)=∣∣∣∣111−31−2713∣∣∣∣det(A)=| 1−371111−23 |

Q&A

Can we use the same method to find the determinant of alarger matrix?

No, this method only worksfor 2×22×2 and 3×33×3 matrices. For largermatrices it is best to use a graphing utility or computersoftware.

Using Cramer’s Rule to Solve a Systemof Three Equations in Three Variables

Now that we can find the determinant of a 3 × 3matrix, we can apply Cramer’s Rule to solve a systemof three equations in three variables. Cramer’s Rule isstraightforward, following a pattern consistent with Cramer’s Rulefor 2 × 2 matrices. As the order of the matrix increases to 3 × 3,however, there are many more calculations required.

When we calculate the determinant to be zero, Cramer’s Rulegives no indication as to whether the system has no solution or aninfinite number of solutions. To find out, we have to performelimination on the system.

Consider a 3 × 3 system of equations.

x=DxD,y=DyD,z=DzD,D≠0x=DxD,y=DyD,z=DzD,D≠0

where

If we are writing the determinant Dx,Dx, we replacethe xx column with the constant column. If we are writingthe determinant Dy,Dy, we replace the yy columnwith the constant column. If we are writing thedeterminant Dz,Dz, we replace the zz columnwith the constant column. Always check the answer.

EXAMPLE 4

Solving a 3 × 3 System Using Cramer’sRule

Find the solution to the given 3 × 3 system using Cramer’sRule.

x+y−z=63x−2y+z=−5x+3y−2z=14x+y−z=63x−2y+z=−5x+3y−2z=14

Answer

TRYIT #3

Use Cramer’s Rule to solve the 3 × 3 matrix.

x−3y+7z=13x+y+z=1x−2y+3z=4x−3y+7z=13x+y+z=1x−2y+3z=4

EXAMPLE 5

Using Cramer’s Rule to Solve anInconsistent System

Solve the system of equations using Cramer’s Rule.

3x−2y=4 (1)6x−4y=0 (2)3x−2y=4 (1)6x−4y=0 (2)

Answer

9.9: Solving Systems with Cramer's Rule (6)

EXAMPLE 6

Use Cramer’s Rule to Solve aDependent System

Solve the system with an infinite number of solutions.

x−2y+3z=03x+y−2z=02x−4y+6z=0(1)(2)(3)x−2y+3z=0(1)3x+y−2z=0(2)2x−4y+6z=0(3)

Answer

9.9: Solving Systems with Cramer's Rule (7)

Understanding Properties ofDeterminants

There are many properties of determinants. Listed here aresome properties that may be helpful in calculating the determinantof a matrix.

PROPERTIES OFDETERMINANTS

If the matrix is in upper triangular form, the determinantequals the product of entries down the main diagonal.
When two rows are interchanged, the determinant changessign.
If either two rows or two columns are identical, thedeterminant equals zero.
If a matrix contains either a row of zeros or a column ofzeros, the determinant equals zero.
The determinant of an inverse matrix A−1A−1 is thereciprocal of the determinant of the matrix A.A.
If any row or column is multiplied by a constant, thedeterminant is multiplied by the same factor.

EXAMPLE 7

Illustrating Properties ofDeterminants

Illustrate each of the properties of determinants.

Answer

EXAMPLE 8

Using Cramer’s Rule and DeterminantProperties to Solve a System

Find the solution to the given 3 × 3 system.

2x+4y+4z=23x+7y+7z=−5 x+2y+2z=4(1)(2)(3)2x+4y+4z=2(1)3x+7y+7z=−5(2) x+2y+2z=4(3)

Answer

MEDIA

Access these online resources for additional instruction andpractice with Cramer’s Rule.

9.8 SectionExercises

Verbal

Explain why we can always evaluate the determinant of a squarematrix.

Examining Cramer’s Rule, explain why there is no unique solutionto the system when the determinant of your matrix is 0. Forsimplicity, use a 2×22×2 matrix.

Explain what it means in terms of an inverse for a matrix tohave a 0 determinant.

The determinant of 2×22×2 matrix AA is 3. Ifyou switch the rows and multiply the first row by 6 and the secondrow by 2, explain how to find the determinant and provide theanswer.

Algebraic

For the following exercises, find the determinant.

∣∣∣1324∣∣∣| 1234 |

∣∣∣−132−4∣∣∣| −123−4 |

∣∣∣2−1−56∣∣∣| 2−5−16 |

∣∣∣−8−145∣∣∣| −84−15 |

∣∣∣130−4∣∣∣| 103−4 |

10.

∣∣∣10020−10∣∣∣| 10200−10 |

Technology

For the following exercises, use the determinant function on agraphing utility.

45.

∣∣∣∣∣∣1010020281349003∣∣∣∣∣∣| 1089021010300243 |

46.

∣∣∣∣∣∣10300−90121−2113−1−2∣∣∣∣∣∣| 10210−91330−2−1011−2 |

47.

∣∣∣∣∣∣∣1200011200710020452,0002∣∣∣∣∣∣∣| 1217401210050022,0000002 |

48.

∣∣∣∣∣∣1247035800690000∣∣∣∣∣∣| 1000230045607890 |

Real-World Applications

For the following exercises, create a system of linear equationsto describe the behavior. Then, calculate the determinant. Willthere be a unique solution? If so, find the unique solution.

49.

Two numbers add up to 56. One number is 20 less than theother.

50.

Two numbers add up to 104. If you add two times the first numberplus two times the second number, your total is 208

51.

Three numbers add up to 106. The first number is 3 less than thesecond number. The third number is 4 more than the firstnumber.

52.

Three numbers add to 216. The sum of the first two numbers is112. The third number is 8 less than the first two numberscombined.

For the following exercises, create a system of linear equationsto describe the behavior. Then, solve the system for all solutionsusing Cramer’s Rule.

53.

You invest $10,000 into two accounts, which receive 8% interestand 5% interest. At the end of a year, you had $10,710 in yourcombined accounts. How much was invested in each account?

54.

You invest $80,000 into two accounts, $22,000 in one account,and $58,000 in the other account. At the end of one year, assumingsimple interest, you have earned $2,470 in interest. The secondaccount receives half a percent less than twice the interest on thefirst account. What are the interest rates for your accounts?

55.

A movie theater needs to know how many adult tickets andchildren tickets were sold out of the 1,200 total tickets. Ifchildren’s tickets are $5.95, adult tickets are $11.15, and thetotal amount of revenue was $12,756, how many children’s ticketsand adult tickets were sold?

56.

A concert venue sells single tickets for $40 each and couple’stickets for $65. If the total revenue was $18,090 and the 321tickets were sold, how many single tickets and how many couple’stickets were sold?

57.

You decide to paint your kitchen green. You create the color ofpaint by mixing yellow and blue paints. You cannot remember howmany gallons of each color went into your mix, but you know therewere 10 gal total. Additionally, you kept your receipt, and knowthe total amount spent was $29.50. If each gallon of yellow costs$2.59, and each gallon of blue costs $3.19, how many gallons ofeach color go into your green mix?

58.

You sold two types of scarves at a farmers’ market and wouldlike to know which one was more popular. The total number ofscarves sold was 56, the yellow scarf cost $10, and the purplescarf cost $11. If you had total revenue of $583, how many yellowscarves and how many purple scarves were sold?

59.

Your garden produced two types of tomatoes, one green and onered. The red weigh 10 oz, and the green weigh 4 oz. You have 30tomatoes, and a total weight of 13 lb, 14 oz. How many of each typeof tomato do you have?

60.

At a market, the three most popular vegetables make up 53% ofvegetable sales. Corn has 4% higher sales than broccoli, which has5% more sales than onions. What percentage does each vegetable havein the market share?

61.

At the same market, the three most popular fruits make up 37% ofthe total fruit sold. Strawberries sell twice as much as oranges,and kiwis sell one more percentage point than oranges. For eachfruit, find the percentage of total fruit sold.

62.

Three bands performed at a concert venue. The first band charged$15 per ticket, the second band charged $45 per ticket, and thefinal band charged $22 per ticket. There were 510 tickets sold, fora total of $12,700. If the first band had 40 more audience membersthan the second band, how many tickets were sold for each band?

63.

A movie theatre sold tickets to three movies. The tickets to thefirst movie were $5, the tickets to the second movie were $11, andthe third movie was $12. 100 tickets were sold to the first movie.The total number of tickets sold was 642, for a total revenue of$6,774. How many tickets for each movie were sold?

64.

Men aged 20–29, 30–39, and 40–49 made up 78% of the populationat a prison last year. This year, the same age groups made up82.08% of the population. The 20–29 age group increased by 20%, the30–39 age group increased by 2%, and the 40–49 age group decreasedto 3434 of their previous population. Originally, the30–39 age group had 2% more prisoners than the 20–29 age group.Determine the prison population percentage for each age group lastyear.

65.

At a women’s prison down the road, the total number of inmatesaged 20–49 totaled 5,525. This year, the 20–29 age group increasedby 10%, the 30–39 age group decreased by 20%, and the 40–49 agegroup doubled. There are now 6,040 prisoners. Originally, therewere 500 more in the 30–39 age group than the 20–29 age group.Determine the prison population for each age group last year.

For the following exercises, use this scenario: Ahealth-conscious company decides to make a trail mix out ofalmonds, dried cranberries, and chocolate-covered cashews. Thenutritional information for these items is shownin Table1.

	Fat (g)	Protein (g)	Carbohydrates (g)
Almonds (10)	6	2	3
Cranberries (10)	0.02	0	8
Cashews (10)	7	3.5	5.5

Table 1

66.

For the special “low-carb”trail mix, there are 1,000 pieces ofmix. The total number of carbohydrates is 425 g, and the totalamount of fat is 570.2 g. If there are 200 more pieces of cashewsthan cranberries, how many of each item is in the trail mix?

67.

For the “hiking” mix, there are 1,000 pieces in the mix,containing 390.8 g of fat, and 165 g of protein. If there is thesame amount of almonds as cashews, how many of each item is in thetrail mix?

68.

For the “energy-booster” mix, there are 1,000 pieces in the mix,containing 145 g of protein and 625 g of carbohydrates. If thenumber of almonds and cashews summed together is equivalent to theamount of cranberries, how many of each item is in the trailmix?