If x and y satisfy the system of equations ⁻¹ x + ⁻¹ y = 2 3 ⁻¹ x - ⁻¹ y = 3 and k = 6y + 18x , then the value of ⁻¹( k) + ⁻¹( k) is

Step-by-step solution

Using the identity ⁻¹ = 2 - ⁻¹ , the second equation becomes: ( 2 - ⁻¹ x ) - ( 2 - ⁻¹ y ) = 3 ⁻¹ y - ⁻¹ x = 3 We now have a linear system in terms of ⁻¹ x and ⁻¹ y : ⁻¹ y + ⁻¹ x = 2 3 ⁻¹ y - ⁻¹ x = 3 Adding the two equations gives: 2 ⁻¹ y = ⁻¹ y = 2 y = 1 Subtracting the second equation from the first gives: 2 ⁻¹ x = 3 ⁻¹ x = 6 x = 1 2 Now, calculate k : k = 6(1) + 18 ( 1 2 ) = 6 + 9 = 15 We need to evaluate ⁻¹( 15) + ⁻¹( 15) . For ⁻¹( 15) : Since 15 radians lies in the interval [4 , 5 ] (as 4 12.56 and 5 15.70 ),