Chapter Nine — Matrix Operations and Calculations

The Humongous Book of Algebra Problems

200

Cramer’s Rule

Double-decker matrices that solve systems

9.38 According to Cramer’s Rule, the solution to a system of two linear equations in

two variables is . Given the system below, identify matrices C,

X, and Y.

Matrix C consists of the system’s coefﬁcients. The x-coefﬁcients comprise the

ﬁrst column of matrix C and the y-coefﬁcients comprise the second.

The remaining matrices, X and Y, are created by replacing individual columns

of C with the column of constants: .

To generate matrix X, replace the ﬁrst column of C with the column of

constants.

To generate matrix Y, replace the second column of C with the column of

constants.

Note: Problems 9.39–9.41 refer to the system of equations below.

9.39 Use variable elimination to solve the system.

Multiply the ﬁrst equation by 2 and multiply the second equation by –3.

Add the equations of the modiﬁed system and solve for x.

The constants

are the numbers

with no variables next

to them—usually found

on the right side of

the equal sign.

To get

the X matrix,

replace the x-

coefcients in C

with the numbers

from across the equal

sign. To get the Y

matrix, replace the

y-coefcients

instead.

To review

variable elimination,

check out Problems

8.19–8.28.

Chapter Nine — Matrix Operations and Calculations

The Humongous Book of Algebra Problems

201

Substitute x = 6 into either equation of the original system to calculate the

corresponding value of y.

The solution to the system is (x,y) = (6,–7).

Note: Problems 9.39–9.41 refer to the system of equations below.

9.40 Construct the matrices C, X, and Y that are required to solve the system using

Cramer’s Rule.

According to Problem 9.38, the ﬁrst column of matrix C consists of the system’s

x-coefﬁcients and the second column consists of the y-coefﬁcients.

To generate matrix X, replace the column of x-coefﬁcients with the constants

from the system.

Similarly, generate Y by replacing the column of y-coefﬁcients with the column

of constants.

Chapter Nine — Matrix Operations and Calculations

The Humongous Book of Algebra Problems

202

Note: Problems 9.39–9.41 refer to the system of equations below.

9.41 Use Cramer’s Rule to verify the solution to the system generated in Problem

9.39.

According to problem 9.40, , , and .

Calculate the determinant of each matrix.

Substitute , , and into the Cramer’s Rule formula to

calculate the solution to the system.

9.42 According to Problem 7.12, the solution to the system below is . Use

Cramer’s Rule to verify the solution.

Construct matrices C, X, and Y, as directed by Problem 9.40.

Calculate the determinants of the matrices.

Apply the Cramer’s Rule formula to calculate the solution to the system.

The constants

in this system

are 1 and 17.

They replace the x-

coefcients (6 and 14)

in the X matrix and

the y-coefcients

(1 and –5) in the Y

matrix.

Chapter Nine — Matrix Operations and Calculations

The Humongous Book of Algebra Problems

203

9.43 Solve the system of equations below using Cramer’s Rule.

Before constructing the matrices required by Cramer’s Rule, rewrite the second

equation of the system in standard form; move the x-term left of the equal sign

and multiply each term by 5 to eliminate the fractions.

The modiﬁed system now contains two linear equations in standard form.

Construct matrices C, X, and Y required by Cramer’s Rule and calculate the

determinant of each.

Apply the Cramer’s Rule formula to calculate the solution to the system.

Line up

the x’s, y’s, and

constants of the

system before you

create any matrices.

The easiest way to get

everything ready for

Cramer’s Rule is to put

the lines in standard

form. (See Problems

6.29–6.36 to review

standard form.)

Chapter Nine — Matrix Operations and Calculations

The Humongous Book of Algebra Problems

204

9.44 Solve the system below using Cramer’s Rule.

Cramer’s Rule is easily modiﬁed to solve systems of three linear equations in

three variables. Instead of three 2 × 2 matrices, four 3 × 3 matrices are required:

C, X, Y, and Z.

Matrix C once again serves as the coefﬁcient matrix—the ﬁrst column contains

the x-coefﬁcients of the system, the second column contains the y-coefﬁcients,

and the third column contains the z-coefﬁcients.

To generate matrices X, Y, and Z, replace the ﬁrst, second, and third columns of

C, respectively, with the column of constants.

Calculate the determinants of all four matrices.

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