Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM.
-A A $ 000 tru trst is to be invested in bonds paying 9%, CDs paying 8%, and mortgages paying 10%. The sum of the amount invested in bonds and the amount invested in CDs must equal the mortgage investment. To earn an $11,400
Annual income from the investments, how much should the bank invest in each?
Let x represent the amount invested in bonds, y the amount invested in CDs, and z the amount invested in mortgages.

Question

Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM.
-A A $  000 tru trst is to be invested in bonds paying 9%, CDs paying 8%, and mortgages paying 10%. The sum of the amount invested in bonds and the amount invested in CDs must equal the mortgage investment. To earn an $11,400
Annual income from the investments, how much should the bank invest in each?
Let x represent the amount invested in bonds, y the amount invested in CDs, and z the amount invested in mortgages.

Quizplus · Accepted Answer

From the problem, we know:

x + y = z (The sum of the amount invested in bonds and the amount invested in CDs must equal the mortgage investment)
0.09x + 0.08y + 0.1z = 11400 (To earn an $11,400 Annual income from the investments)

We have two equations and three variables. However, we can solve for one of the variables in terms of the other two by using the first equation:

z = x + y

Now, we can substitute this into the second equation:

0.09x + 0.08y + 0.1(x + y) = 11400

Simplifying:

0.19x + 0.18y = 11400

This is a linear equation in two variables. We can solve for one of the variables in terms of the other and then substitute back into one of the equations to solve for the other variable. For example, solving for y in terms of x:

0.18y = 11400 - 0.19x

y = (11400 - 0.19x)/0.18

Now, we can substitute this into the first equation:

x + (11400 - 0.19x)/0.18 = 2x + 11400/0.1

Simplifying:

x = 24000/11 ≈ 2181.82

Substituting this value back into the equation for y:

y = (11400 - 0.19(24000/11))/0.18 ≈ 1388.64

Finally, we can use the equation z = x + y to find z:

z = x + y ≈ 3567.45

Therefore, the bank should invest approximately $2181.82 in bonds, $1388.64 in CDs, and $3567.45 in mortgages to earn an $11,400 annual income from the investments.

The only answer choice that matches this solution is C. From the problem, we have:

Amount invested in bonds: x
Amount invested in CDs: y
Amount invested in mortgages: z

It is given that x+y=z and the total amount invested is $11000. Therefore, we have:

x+y+z=11000

To earn an $11,400 Annual income from the investments with the given interest rates, we can set up the following equation:

0.09x+0.08y+0.10z=11400

Substituting z=x+y into the above equation, we get:

0.09x+0.08y+0.10(x+y)=11400
0.19x+0.10y=11400

Simplifying the above equation, we get:

19x+10y=1140000

This is a system of two equations in two variables:

x+y+z=11000
19x+10y=1140000

The solution to this system will give us the values of x, y, and z. However, we are not required to solve the system.

The best choice is C, which gives the correct equations for x, y, and z.

Find the System of Equations to Model the Problem