Answer
$$A) \begin{cases}
a+b& = 150000 \\
0.05a+0.072b &= 9600\\
\end{cases}$$ B) Damian needs to invest $\$ 95,454.55$ and $\$54,545.55$ in each account.
Work Step by Step
Part A
We know that Damian has a total fund of $\$ 150,000$ to invest in two different accounts, say account A and account B. She also wants to a total amount of $\$9,600$ per year from each of her invests at an interest rate of $5\%$ and $7.2\%$ respectively. Let $a$ be the amount invested in account A and $b$ be the amount invested in account B respectively. Then, the sum of the money invested must satisfy the following equation:
$$ a+b = 150000.$$ Similarly, the sum of the interest earned each year must satisfy the following equation: $$ 0.05a+0.072b = 9600.$$Hence, the required system of equation is given by
$$\begin{cases}
a+b& = 150000 \\
0.05a+0.072b &= 9600.
\end{cases}$$ Part B
Solve the above equations for a and b. Multiply the first equation by $-0.05$ and add the result to the second to eliminate $a$. $$\begin{aligned}
-0.05a-0.05b& = -0.05\cdot 150000 \\
-0.05a-0.05b &= -7500.
\end{aligned}$$ Now add.
$$\begin{aligned}
-0.05a-0.05b+0.06a+0.072b &= -7500+9600\\
0.022b&=2100.
\end{aligned}$$ Solve for $b$: $$\begin{aligned}
b & = \frac{2100}{0.022}\\
&\approx \$95,454.55
\end{aligned}$$ Finally, use the first equation to find the value of $a$: $$a= 150000-b = 150000-95454.55= \$54,545.55$$
