You invest a total of $5800 in two investments earning 3.5% and 5.5% simple interest. Your goal is to have a total annual interest income of $283. Write a system of linear equations that represents this situation where x represents the amount invested in the 3.5% fund and y represents the amount invested in the 5.5% fund. Solve this system to determine the smallest amount that you can invest at 5.5% in order to meet your objective.

The annual interest that can be earned through investment of an amount at a simple interest can be calculated through the equation,

I = P x (i)

where I is interest, P is the principal amount, and i is the decimal equivalent of the interest.

Let x be the amount deposited with 3.5% interest. With this representation, the amount deposited with 5.5% is 5800 - x.

The linear equation that would represent the given scenario is,

x (0.035) + (5800 - x) (0.055) = 283

Simplifying the equation,

0.035x + 319 - 0.055x = 283

Combining like terms,

-0.02x = - 36

Dividing by - 0.02,

x = 1800

$5800 - x = $5800 - $1800 = y

y = $4000

Hence, the value that should be deposited to the 5.5% is $40000.