This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Combine the fractions on the left-hand side. The given equation is: (z+1)/(1+i) + (z)/(1-i) = 5+5i To combine the fractions, find a common denominator, which is (1+i)(1-i). (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1+1 = 2 Now rewrite the left-hand side with the common denominator: ((z+1)(1-i))/((1+i)(1-i)) + (z(1+i))/((1-i)(1+i)) = 5+5i ((z+1)(1-i) + z(1+i))/(2) = 5+5i Step 2: Simplify the numerator of the left-hand side. Expand the terms in the numerator: (z+1)(1-i) = z(1) + z(-i) + 1(1) + 1(-i) = z - zi + 1 - i z(1+i) = z + zi Substitute these back into the numerator: (z - zi + 1 - i) + (z + zi) Combine like terms: z - zi + 1 - i + z + zi = (z+z) + (-zi+zi) + 1 - i = 2z + 0 + 1 - i = 2z + 1 - i Step 3: Substitute the simplified numerator back into the equation and solve for z. (2z + 1 - i)/(2) = 5+5i Multiply both sides by 2: 2z + 1 - i = 2(5+5i) 2z + 1 - i = 10+10i Isolate the term with z: 2z = 10+10i - (1-i) 2z = 10+10i - 1 + i Combine the real and imaginary parts on the right-hand side: 2z = (10-1) + (10+1)i 2z = 9 + 11i Divide by 2 to find z: z = (9+11i)/(2) z = (9)/(2) + (11)/(2)i The complex number z in the form a+bi is (9)/(2) + (11)/(2)i.