5) find the complex number z and w such that z+w=4+3i and z-iw=3-2i
subtract them:
w + iw = 4+3i -3 + 2i
w(1+i) = 1 + 5i
w = (1+5i)/(1+i)
"realizing" the denominator
w = (1+5i)/(1+i) * (1-i)/(1-i)
= (1 + 5i - i - 5i^2)/(1 - i^2)
= (6 + 4i)/2 = 3 + 2i
then in z+w = 4+3i
z + 3+2i = 4+3i
z = 1+i
thanks! very clear now
To find the complex numbers z and w that satisfy the given equations, we can employ a two-variable approach using algebraic manipulation.
Let's start by re-arranging the equations:
Equation 1: z + w = 4 + 3i
Equation 2: z - iw = 3 - 2i
Now, let's isolate z in Equation 2 by adding iw to both sides:
z - iw + iw = 3 - 2i + iw
z + 0i = 3 - 2i + iw
z = 3 - 2i + iw
Next, substitute the value of z from Equation 2 into Equation 1:
(3 - 2i + iw) + w = 4 + 3i
Now, we have an equation with only w as the variable. Let's simplify it:
3 - 2i + iw + w = 4 + 3i
Combine like terms:
(3 + w) + (iw + w) = 4 + 3i
Now, group the real terms and the imaginary terms:
(3 + w) + w(i + 1) = 4 + 3i
Since this equation must hold for all complex numbers, the real parts and imaginary parts must be equal on both sides.
Equating the real parts:
3 + w = 4
Solve for w:
w = 4 - 3
w = 1
Equating the imaginary parts:
w(i + 1) = 3i
Expand the left side:
w + wi = 3i
Rearrange the equation:
wi = 3i - w
Since this equation must hold for all complex numbers, the real parts and imaginary parts must be equal on both sides.
Equating the real parts:
0 = 0
This implies that there are no restrictions or constraints on the value of w in this equation. Therefore, w can take any complex value.
Now, substitute the value of w back into Equation 1 to find z:
z = 3 - 2i + iw
z = 3 - 2i + i(1)
Simplify:
z = 3 - 2i + i
z = 3 - 2i + i
z = 3 - i
Thus, the complex numbers that satisfy the given equations are z = 3 - i and w = 1.