###### Example13.5.1

Mohamed has his retirement money invested in two accounts, one "safe" and the other highly speculative. At the beginning of 2017 the two accounts combined had a total of $27,872.00. Over the course of the year the safe account earned 2.5% interest while the speculative account grew by 43%. Mohamed made no deposits to nor withdrawals from either account over the course of the year. At the end of the year the total balance between the two accounts was $35,944.26. How much did Mohamed have invested in each account at the beginning of 2017? Assume that the start of the year the amount in each account was a whole dollar amount.

Let's begin by defining \(x\) to be the amount ($) that Mohamed had invested in the safe account at the beginning of 2017 and \(y\) the amount he had invested in the speculative account at the beginning of 2017. Let's also note that the total growth in the balance of the two accounts over the course of 2017 was $8,072.26 (the difference between $34,966.26 and $27,872.00). The information we know about the accounts is summarized in TableĀ 13.5.2.

Safe Account | Speculative Account | Total | |

Amount at start of 2017 ($) | \(x\) | \(y\) | \(27,872.00\) |

Annual Growth ($) | \(.025x\) | \(.43y\) | \(8,072.26\) |

In each row of the table, the amounts in the safe and speculative accounts sum to to the total amount. That leads to the following system of equations.

We can eliminate \(x\) from the system by multiplying both sides of the first equation by \(-.025\) and then summing the respective sides of the two equations in the resultant system.

Let's equate the sums of the respective sides of the equations and solve the resultant equation for \(y\text{.}\)

Substituting \(18,211\) for \(y\) in the equation \(x+y=27,872\) will enable us to determine the value for \(x\text{.}\)

Let's check the values for \(x\) and \(y\) in the growth equation.

At the beginning of 2017, Mohamed had $9,611.00 invested in the safe account and $18,211.00 invested in the speculative account.