Solve the system of linear equations by Cramer’s rule  3x + y = 19 3x - y = 23

Solution:

Cramer’s rule is used to study the solution Ax = b affected by changes in the entries. For any n × n matrix A and for any b in ℝⁿ, the unique solution x of Ax = b is given by:

x_i = det Aᵢ (b)i = 1, 2, ……, n---------->(1)

det A

The given set of equations that need to be solved using Cramer’s rule are:

3x + y = 19

3x - y  =  23

When we view the system as Ax = b we can write:

A = \(\begin{pmatrix} 3 & 1\\ 3 & -1 \end{pmatrix}\)

A₁ (b) = \(\begin{pmatrix} 19 & 1\\ 23 & -1 \end{pmatrix}\)

A₂(b) = \(\begin{pmatrix} 3 & 19\\ 3 & 23 \end{pmatrix}\)

(Determinant)det A  = (3 × -1)  -  (3 × 1) = -3 - 3 = -6

det A₁ (b)  = (19 × -1) - (23 × 1)  = -19 - 23 = -42

det A₂(b)   = (3 × 23) -  (19 × 3)  = 69 - 57 = 12

x = det A₁ (b)/det A = -42/-6  = 7

y = det  A₂(b) /det A = 12/(-6)  = -2

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Solve the system of linear equations by Cramer’s rule  3x + y = 19 3x - y = 23

Summary:

Solving the system of linear equations 1) 3x + y = 19 and 2) 3x - y = 23 by Cramer’s rule  we get x = 7 and y = -2.