BOHOL ISLAND STATE UNIVERSITY
MAIN CAMPUS – T AGBILARAN CITY
COLLEGE OF ENGINEERING, ARCHITECTURE, AND INDUSTRIAL DESIGN
DEPARTMENT OF CIVIL ENGINEERING
ACTVITY 1F
Solving Linear
Equations Using
Cramer’s Method
MATH 05
NOVEMBER 6, 2023
SUBMITTED BY:
CHRISTINA FAITH P. OCIONES
BSCE 3B
SUBMITTED TO:
ENGR. MAYRELL COQUILLA
INSTRUCTOR
I. Introduction
SOURCE: https://www.youtube.com/watch?app=desktop&v=Ot87qLTODdQ
Cramer's Rule constitutes a mathematical approach that serves as a valuable tool
for resolving systems of linear equations. Its strength particularly shines when
applied to square systems of linear equations, characterized by an equivalence in
the count of equations and variables. The underpinning principle of Cramer's Rule
lies in determinants, which are numerical values intricately tied to matrices and
provide a means for systematically arriving at solutions.
In practical terms, to harness the potential of Cramer's Rule, one must initially
reframe the system of linear equations in a matrix format denoted as Ax = b. In this
representation, A symbolizes the coefficient matrix, while x represents the column
vector housing the variables, and b encompasses the constants expressed in a
column vector format. What sets Cramer's Rule into motion is the determinant of
the coefficient matrix A, succinctly labeled as det(A). If this determinant proves non-
zero, it signals the applicability of Cramer's Rule, indicating the existence of a
distinct, single solution for the system.
Cramer's Rule then unfolds as an endeavor to ascertain the determinants of
matrices that materialize by consecutively substituting each column of A with the
column vector b. The resultant fractions derived from these determinants, divided
by the determinant of the original matrix A, effectively yield the precise values of
the variables in the system. It's worth noting, however, that Cramer's Rule, while
potent in its utility, tends to become computationally demanding as the complexity
of the system increases due to the necessity of calculating multiple determinants,
rendering it less efficient when compared to alternative methods in such scenarios.
II. Problem statement
Use the Cramer’s Rule to find the six unknown values of systems of linear
equations.
6x1 - 2x2 + 5x3 – x4 + 4x5 - 8x6 = 24
2x1 - 6x2 + 3x3 + x4 - 7x5 + 5x6 = 11
4x1 - 2x2 + 6x3 + 5x4 - 7x5 - 3x6 = 19
5x1 - 3x2 + 6x3 - 2x4 - 4x5 + 7x6 = 23
4x1 - 5x2 + 2x3 - 6x4 + x5 - 8x6 = 18
2x1 - 4x2 + x3 - 3x4 + 6x5 - 7x6 = 12
III. Methodology/Algorithm
The following algorithm shows how to locate the root(s) of the equation using false-
position method.
Step 1. Start
Step 2. Create a matrix system of the following linear equations.
Step 3. Calculate the determinant of the matrix coefficients (D, D 1, D2, D3, D4, D5,
D6)
Step 4. Solve for x1, x2, x3, x4, x5, and x6 using this equation.
Step 5. End.
IV. Algorithm Flow Chart
STAR
Matrix System of Linear
Equations
Determina
nts
EN
V. Solving Using Cramer’s Rule
on Excel
Solving Linear Equations with 6 unknowns using Cramer’s Rule on Excel.
A b
6 -2 5 -1 4 -8 24
2 -6 3 1 -7 5 11
4 -2 6 5 -7 -3 19
5 3 6 -2 -4 7 23
4 -5 2 -6 1 -8 18
2 -4 1 -3 6 -7 12
D
6 -2 5 -1 4 -8
2 -6 3 1 -7 5
4 -2 6 5 -7 -3
DETERMINATE: 39871
5 3 6 -2 -4 7
4 -5 2 -6 1 -8
2 -4 1 -3 6 -7
D1
24 -2 5 -1 4 -8
11 -6 3 1 -7 5
19 -2 6 5 -7 -3 D1: -41704
23 3 6 -2 -4 7
18 -5 2 -6 1 -8
12 -4 1 -3 6 -7
D2
6 24 5 -1 4 -8
2 11 3 1 -7 5
4 19 6 5 -7 -3 D2: -17769
5 23 6 -2 -4 7
4 18 2 -6 1 -8
2 12 1 -3 6 -7
D3
6 -2 24 -1 4 -8
2 -6 11 1 -7 5
4 -2 19 5 -7 -3 D3: 198860
5 3 23 -2 -4 7
4 -5 18 -6 1 -8
2 -4 12 -3 6 -7
D4
6 -2 5 24 4 -8
2 -6 3 11 -7 5
4 -2 6 19 -7 -3 D4: -50966
5 3 6 23 -4 7
4 -5 2 18 1 -8
2 -4 1 12 6 -7
D5
6 -2 5 -1 24 -8
2 -6 3 1 11 5
4 -2 6 5 19 -3 D5: 11397
5 3 6 -2 23 7
4 -5 2 -6 18 -8
2 -4 1 -3 12 -7
D6
6 -2 5 -1 4 24
2 -6 3 1 -7 11
4 -2 6 5 -7 19 D6: -10092
5 3 6 -2 -4 23
4 -5 2 -6 1 18
2 -4 1 -3 6 12
VI. Results
The unknown values of the linear equations are:
X1 = -1.04597
X2 = -0.44566
X3 = 4.987585
X4 = -1.27827
X5 = 0.285847
X6 = -0.25312
