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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