1.6 Solution of Simultaneous Linear Equations with Complex Coefficients and Exercise 1.4

Example 21

Solve the given simultaneous linear equations with complex coefficients for $z$ and $w$:

$5z - (3 + i)w = 7 - i$

$(2 - i)z + 2i w = 4$

Solution

$5z - (3 + i)w = 7 - i$ ... (i)

$(2 - i)z + 2i w = 4$ ... (ii)

Step 1: From equation (i), we solve for $w$ in terms of $z$.

$5z - (3 + i)w = 7 - i$

$(3 + i)w = 5z - 7 + i$

$w=\frac{5z-7+i}{3+i}$$w = \frac{5z - 7 + i}{3 + i}$ ... (iii)

Step 2: Put the expression of $w$ in equation (ii)

$(2-i)z+2i\left(\frac{5z-7+i}{3+i}\right)=4$$(2 - i)z + 2i\left(\frac{5z - 7 + i}{3 + i}\right) = 4$

$(2 - i)(3 + i)z + 2i(5z - 7 + i) = 4(3 + i)$

$(6 + 2i - 3i - i^2)z + 10iz - 14i + 2i^2 = 12 + 4i$

$(6 - i + 1)z + 10iz - 14i - 2 = 12 + 4i$

$(7 - i)z + 10iz - 14i - 2 = 12 + 4i$

$7z - iz + 10iz - 14i - 2 = 12 + 4i$

$7z + 9iz = 12 + 4i + 14i + 2$

$7z + 9iz = 14 + 18i$

$z(7 + 9i) = 2(7 + 9i)$

$z=\frac{2(7+9i)}{7+9i}$$z = \frac{2(7 + 9i)}{7 + 9i}$

$z = 2$