Answer
$a = \frac{28}{19}$
$b = \frac{25}{19}$
Work Step by Step
$(5+3i)a+(2+5i)b = 10 + 11i$
Expand the above expression:
$5a + 3ia + 2b + 5ib = 10 + 11i$
Combine like terms:
$(5a+2b) + (3a+5b)i = 10 + 11i$
In order for the complex number to equal $10 + 11i$, the real component must equal $10$ and the imaginary component must equal $11$.
Equation 1: This is an expression from the real component of the complex number.
$5a + 2b = 10$
Equation 2: This is an expression from the imaginary component of the complex number.
$3a + 5b = 11$
Solve the above system of equations using a matrix:
Form a matrix from the two equations above:
$\begin{bmatrix}
5 & 2 & |10 \\
3 & 5 & |11\\
\end{bmatrix}$
Simplify the matrix to a form where it would be easy to solve for a and b:
$\begin{bmatrix}
5 & 2 & |10 \\
3 & 5 & |11\\
\end{bmatrix}$ ~ $\begin{bmatrix}
5 & 2 & |10 \\
0 & 19 & |25\\
\end{bmatrix}$
From the simplified matrix, there are two equations:
Equation 3: This is from the first row of the matrix.
$5a + 2b = 10$
Equation 4: This is from the second row of the matrix.
$19b = 25$
Solve Equation 4, to determine the value of b:
$b = \frac{25}{19}$
Substitute the value of b into Equation 3 to solve for the value of a:
$5a + 2(\frac{25}{19}) = 10$
$5a + \frac{50}{19} = 10$
$5a = 10 - \frac{50}{19}$
$5a = \frac{140}{19}$
$a = \frac{28}{19}$
In summary:
$a = \frac{28}{19}$
$b = \frac{25}{19}$
