Answer
$\$47,000$ in stocks
$\$23,000$ in bonds
Work Step by Step
We know that Mike invested $\$70,000 $ in both stocks and bonds with a return of $\$6305$ in the first year. Let $ x$ be the amount invested in stock at the interest rate of $9.5\%= 0.095$ and $ y$ be the amount invested in bonds at the interest rate of $8\%= 0.08$.Then, the sum of the money invested must satisfy the following equation: $$ x+y = 70000.$$ Similarly, the sum of the interest earned each year must satisfy the following equation: $$ 0.095x+0.08y = 6305. $$ We can now solve the system of equations to figure out the amount of money that must be invested in both stocks and bonds. $$\begin{cases}
x+y& = 70000 \\
0.095x+0.08y &= 6305.
\end{cases}$$ Solve the above equations for $x$ and $y$. Multiply the first equation by $-0.095$ and add the result to the second to eliminate $x$. $$\begin{cases}
-0.095\cdot (x+y)& = -0.095\cdot 70000 \\
-0.095x-0.08y &= -6650\\
\end{cases}$$ Now add this to the second equation and solve for $y$. $$\begin{cases}
-0.095x-0.095y &= -6650\\
0.095x+0.08y &= 6305.
\end{cases}$$ Rearrange: $$\begin{aligned}
(0.095x-0.095x)+(0.08x-0.095y)&=6305-6650\\
-0.015y&= -345\\
y & = \frac{-345}{-0.015}\\
&= \$23,000.
\end{aligned}$$ Finally, use the first equation to find the value of $x$. $$x= 70000-y = 70000-23000= \$47,000.$$ He needs to invest $\$47,000$ in stocks and $\$23,000$ in bonds, respectively.
