Carletta has $10,000 to invest. As her financial consultant you recommend that she invest in Treasury Bills that yield 6%, Treasury Bonds that yield 7%, and coroporate bonds that yield 8%. Carletta wants to have an annual income of $680, and the amount invested in corporate bonds must be half that invested in Treasury Bills. Find the equations.

Bills + Bonds + Corporate=10000

Bills + Bonds + 1/2 Bills=10000
That is equation 1)
Bills(.06)+ bonds(.07) + corporate(.08)=680
Bills*.06 + bonds*.07 +1/2 bills*.08=680 or equation 2)
1.5*bills(.06) + bonds(.07)=680

So now you have two equations, two unknowns, and you can solve.

To solve this problem, we need to set up two equations based on the given information. Let's start by assigning variables to the unknown quantities.

Let x be the amount invested in Treasury Bills.
Then, 2x represents the amount invested in corporate bonds.

First Equation:
The annual income Carletta wants to earn is $680. We can represent this by the sum of the incomes from each type of investment:

0.06x + 0.07(0.5x) + 0.08(2x) = 680

In this equation:
0.06x represents the income from Treasury Bills (yielding 6%).
0.07(0.5x) represents the income from corporate bonds (yielding 7%).
0.08(2x) represents the income from Treasury Bonds (yielding 8%).

Simplifying the equation gives us:
0.06x + 0.035x + 0.16x = 680
0.255x = 680

Second Equation:
The total amount invested should be $10,000, so we can write:

x + 2x + 0.5x = 10,000

Simplifying the equation gives us:
3.5x = 10,000

Now we have a system of equations:
0.255x = 680
3.5x = 10,000

To find the solution for x (the amount invested in Treasury Bills), we will solve this system of equations using algebraic methods, such as substitution or elimination.