Question

Linear

algebra
solve all please
5. Use Cramers rule to compute the solution of the following linear systems of equations (a) \[ \begin{array}{l} 5 x_{1}+7 x_{2}=3 \\ 2 x_{1}+4 x_{2}=1 \end{array} \] (b) \[ \begin{aligned} 2 x_{1}+x_

Answer 1

To solve the system of linear equations using Cramer's Rule, find the determinants of the coefficient matrix and the individual variable matrices, then calculate the solution by dividing the determinants.

Step-by-step explanation:

To solve the system of linear equations using Cramer's Rule, we need to find the determinants of the coefficient matrix and the individual variable matrices.

Solution for (a)

1. Calculate the determinant of the coefficient matrix:

• Determinant of the coefficient matrix = ad - bc
• Determinant of (5 7, 2 4) = (5 * 4) - (7 * 2) = 20 - 14 = 6

Calculate the determinant of the x1 matrix:

• Replace the first column (coefficient of x1) with the constant column: (3, 1)
• Determinant of the x1 matrix = (3 * 4) - (1 * 7) = 12 - 7 = 5

Calculate the determinant of the x2 matrix:

• Replace the second column (coefficient of x2) with the constant column: (5, 2)
• Determinant of the x2 matrix = (5 * 1) - (2 * 3) = 5 - 6 = -1

Calculate the solution:

• x1 = determinant of x1 matrix / determinant of coefficient matrix = 5 / 6
• x2 = determinant of x2 matrix / determinant of coefficient matrix = -1 / 6

Therefore, the solution for (a) is x1 = 5/6 and x2 = -1/6.