Math Problem Statement
solve using cramers rule
Solve the system of equations using Cramer's Rule:
2x - y + z = 1
x + 2y - 4z = 3
4x + 3y - 7z = -8
Solution
The system of equations provided is:
2x - y + z &= 1 \\ x + 2y - 4z &= 3 \\ 4x + 3y - 7z &= -8 \end{aligned}$$ We will solve this system using Cramer's Rule. To apply Cramer's Rule, we need to calculate the determinant of the coefficient matrix and the determinants of matrices formed by replacing each column of the coefficient matrix with the constants on the right-hand side. ### Step 1: Write the system as a matrix equation $$A \cdot X = B$$ The system can be written as: $$A = \begin{bmatrix} 2 & -1 & 1 \\ 1 & 2 & -4 \\ 4 & 3 & -7 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ 3 \\ -8 \end{bmatrix}$$ ### Step 2: Find the determinant of the coefficient matrix $$A$$ $$\text{det}(A) = \begin{vmatrix} 2 & -1 & 1 \\ 1 & 2 & -4 \\ 4 & 3 & -7 \end{vmatrix}$$ We calculate this determinant. ### Step 3: Replace each column of $$A$$ with the constants and compute the determinants. - $$A_x$$ (replace the first column of $$A$$ with $$B$$): \[ A_x = \beginAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Determinants
Formulas
Cramer's Rule: x = det(A_x)/det(A), y = det(A_y)/det(A), z = det(A_z)/det(A)
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Cramer's Rule
Suitable Grade Level
Grades 11-12 or College Level
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