Answer
$a = 3$
$b = 1$
Work Step by Step
Note: $i^{2} = \sqrt -1\sqrt -1 = -1$
$\frac{5+ai}{8-2i} = \frac{b+4i}{5+3i}$
Cross multiply the above expression:
$(5+ai)(5+3i) = (b+4i)(8-2i)$
Expand the above expression:
$25 + 15i + 5ai + 3ai^{2} = 8b - 2ib + 32i - 8i^{2}$
As seen in the note above, $i^{2} = -1$, so substitute in $-1$.
$25 + 15i + 5ai + 3a(-1) = 8b - 2ib + 32i - 8(-1)$
Simplify:
$25 + 15i + 5ai - 3a = 8b - 2ib + 32i + 8$
$25 + 15i + 5ai - 3a - 8b + 2ib - 32i - 8 = 0$
Combine like terms:
$17 - 17i + (5a+2b)i + (-3a-8b) = 0$
Rewrite the above expression:
$(-3a-8b) + (5a+2b)i = -17 + 17i$
In order for the complex number to equal $-17 + 17i$, the real component must equal $-17$ and the imaginary component must equal $17$.
Equation 1: This is an expression from the real component of the complex number.
$-3a-8b = -17$
Equation 2: This is an expression from the imaginary component of the complex number.
$5a + 2b = 17$
Solve the above system of equations using a matrix:
Form a matrix from the two equations above:
$\begin{bmatrix}
-3 & -8 & |-17 \\
5 & 2 & |17\\
\end{bmatrix}$
Simplify the matrix to a form where it would be easy to solve for a and b:
$\begin{bmatrix}
-3 & -8 & |-17 \\
5 & 2 & |17\\
\end{bmatrix}$ ~ $\begin{bmatrix}
-3 & -8 & |-17 \\
0 & -34 & |-34\\
\end{bmatrix}$
From the simplified matrix, there are two equations:
Equation 3: This is from the first row of the matrix.
$-3a-8b = -17$
Equation 4: This is from the second row of the matrix.
$-34b = -34$
Solve Equation 4, to determine the value of b:
$b = 1$
Substitute the value of b into Equation 3 to solve for the value of a:
$-3a-8(1) = -17$
$-3a-8 = -17$
$-3a = -17 + 8$
$-3a = -9$
$a = 3$
In summary:
$a = 3$
$b = 1$
