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Consider the following system of equations: 2x₁ - x₂ = 30 -2x₁ - 2x₂ + x₃ = 50 x₂ + 3x₃ = 140 a. Write the system in matrix form Ax=b and identify the matrices A, x, and b. b. Find the determinant of matrix A and determine whether the system of equations has a unique solution, no solution, or infinitely many solutions based on the determinant value. c. Use the method of your choice (Gaussian elimination, Cramer's rule, etc.) to solve for the values of x₁, x₂, and x₃ in the system of equations. d. Verify your solution by substituting the values you found into the original equations to ensure they satisfy all the equations.

User Ashwel
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Final answer:

a. The system of equations can be written in matrix form Ax=b as A = [[2, -1, 0], [-2, -2, 1], [0, 1, 3]], x = [[x1], [x2], [x3]], and b = [[30], [50], [140]]. b. The determinant of matrix A is 7, indicating that the system of equations has a unique solution. c. By using Gaussian elimination, we find the solution to be x1 = 10, x2 = -10, and x3 = 50. d. The solution can be verified by substituting the values into the original equations.

Step-by-step explanation:

a. To write the system of equations in matrix form Ax=b, we can arrange the coefficients of the variables in a matrix A, the variables themselves in a matrix x, and the constants on the right-hand side of the equations in a matrix b. The matrices A, x, and b can be defined as follows:

A = [[2, -1, 0], [-2, -2, 1], [0, 1, 3]]
x = [[x1], [x2], [x3]]
b = [[30], [50], [140]]

b. To find the determinant of matrix A, we can use the formula for a 3x3 matrix:

det(A) = 2(3) - (-1)(1) + 0(1) = 6 + 1 + 0 = 7

Since the determinant is non-zero (7 ≠ 0), the system of equations has a unique solution.

c. To solve the system of equations, we can use Gaussian elimination:

Step 1: Add 2 times the first equation to the second equation.
Step 2: Subtract the second equation from 3 times the third equation.
Step 3: Solve the resulting system of equations.

The solution is x1 = 10, x2 = -10, and x3 = 50.

d. To verify the solution, we substitute the values of x1, x2, and x3 into the original equations:

Equation 1: 2(10) - (-10) = 30
Equation 2: -2(10) - 2(-10) + (50) = 50
Equation 3: (-10) + 3(50) = 140

These equations are all true, so the solution is correct.

User Onlyme
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