Answer
$\$93,750$ and $\$81,250$
Work Step by Step
Part A
We know that Joan has $\$175,000 $ to invest in two different accounts, say account A and account B. She also wants to earn a total amount of $\$12,000$ per year from each of her investment at an interest rate of $9\%$ and $5\%$ 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: $$ a+b = 175000.$$ Similarly, the sum of the interest earned each year must satisfy the following equation: $$ 0.09a+0.05b = 12000.$$ We can now solve the system of equations to figure out the amount of money that she should invest in each account. $$\begin{aligned}
a+b& = 175000 \\
0.09a+0.05b &= 12000.
\end{aligned}$$ Solve the above equations for $a$ and $b$. Multiply the first equation by $-0.09$ and add the result to the second to eliminate $a$. $$\begin{aligned}
-0.09\cdot (a+b)& = -0.09\cdot 175000 \\
-0.09a-0.09b &= -15750.
\end{aligned}$$ Now add this to the second equation and solve for $b$. $$\begin{aligned}
-0.09a-0.09b &= -15750\\
0.09a+0.05b &= 12000.
\end{aligned}$$ Rearrange: $$\begin{aligned}
-0.04b&= - 3750\\
b & = \frac{3750}{0.04}\\
&= \$93,750.
\end{aligned}$$ Finally, use the first equation to find the value of $a$. $$a= 175000-b = 93750-800000= \$81,250.$$ She needs to invest $\$93,750$ and $\$81,250$ respectively.
