Answer
$\$360,000$ at $5\%$
$\$240,000$ at $6.5\%$
Work Step by Step
We know that Nikki has $\$600,000 $ to invest in two different accounts, say account A and account B. She also wants to earn a total amount of $\$33,600$ per year from her investments at an interest rate of $5\%$ and $6.5\%$ respectively. Let $a$ be the amount invested in account A and $b$ be the amount invested in account B. Then, the sum of the money invested must satisfy the following equation: $$ a+b = 600000.$$ Similarly, the sum of the interest earned each year must satisfy the following equation: $$ 0.05a+0.065b = 33600.$$ We can now solve the system of equations to figure out the amount of money that she should invest in each account. $$\begin{aligned}
a+b& = 600000 \\
0.05a+0.065b &= 33600.
\end{aligned}$$ Solve the above equations for $a$ and $b$. Multiply the first equation by $-0.05$ and add the result to the second to eliminate $a$. $$\begin{aligned}
-0.05\cdot (a+b)& = -0.05\cdot 600000 \\
-0.05a-0.05b &= -30000.
\end{aligned}$$ Now add this to the second equation and solve for $b$. $$\begin{aligned}
-0.05a-0.05b &= -30000\\
0.05a+0.065b &= 33600.
\end{aligned}$$ Rearrange: $$\begin{aligned}
(0.05a-0.05a)+(0.065b-0.05b)&=33600-30000)\\
0.015b&= 3600\\
b & = \frac{3600}{0.015}\\
&= \$240,000.
\end{aligned}$$ Finally, use the first equation to find the value of $a$. $$a= 600000-b = 600000-240000= \$360,000.$$ She needs to invest $\$360,000$ at the $5\%$ interest rate and $\$240,000$ at the $6.5\%$ respectively.
