Math Problem Statement
Solution
To solve this system of equations using Cramer's Rule, we first need to identify the system of equations and set up matrices.
Given System of Equations
The equations are:
Since Cramer's Rule requires a square matrix (a system with the same number of equations as variables), we first rewrite this system by adding a term for in the third equation with a coefficient of 0:
Step 1: Coefficient Matrix (Matrix )
The coefficient matrix is:
Step 2: Determinant of (denoted )
To use Cramer's Rule, we need to calculate the determinant of .
Expanding along the first row:
Calculating each minor:
-
For :
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For :
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For :
Putting it all together:
Step 3: Calculating Determinants for , , and
Now, we'll create matrices , , and by replacing the columns of with the constants from the right-hand side of the equations , and then calculate their determinants.
Determinant for ()
Replace the first column of with :
can be calculated similarly, but I will outline the steps briefly if you’d like. With , , and , Cramer's Rule gives us:
Would you like me to proceed with these calculations in detail?
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Cramer's Rule
Systems of Equations
Determinants
Formulas
Cramer's Rule formula: x = det(A_x)/det(A), y = det(A_y)/det(A), z = det(A_z)/det(A)
Determinant calculation for a 3x3 matrix
Theorems
Cramer's Rule
Suitable Grade Level
Grades 11-12 or College level
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