Question 1072150: Solve using system of linear equations.
A $70,000 trust is to be invested in bonds paying 9%, CDs paying 7%, and mortgages paying 10%. The bond and CD investment together must equal the mortgage investment. To earn a $6430 annual income from the investments, how much should the bank invest in bonds?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let:
a = investment in bonds.
b = investment in CDs.
c = investment in mortgages.
total investment is equal to 70,000.
a + b + c = 70,000
total interest is equal to 6,430.
.09 * a + .07 * b + .10 * c = 6,430.
investment in bonds and CDs must be equal to investment in mortgages.
a + b = c
you have 2 equations that needs to be solved simultaneously.
they are:
a + b + c = 70,000
.09 * a + .07 * b + .10 * c = 6,430.
you are given that a + b = c
you can use this information to reduce the number of unknown variables that have to be solved.
replace c with a + b in both equations and you get:
a + b + c = 70,000 becomes a + b + a + b = 70,000 which becomes 2 * a + 2 * b = 70,000.
.09 * a + .07 * b + .10 * c = 6,430 becomes .09 * a + .07 * b + .10 * (a + b) = 6,430 which becomes .09 * a + .07 * b + .10 * a + .10 * b = 6,430 which becomes .19 * a + .19 * b = 6,430.
your 2 equations now become:
2 * a + 2 * b = 70,000
.19 * a + .17 * b = 6,430
in the first equation, solve for a to get:
a = 35,000 -b
in the second equation, replace a with 35,000 - b to get:
.19 * (35,000 - b) + .17 * b = 6,430.
simplify to get:
6,650 - .19 * b + .17 * b = 6,430.
combine like terms to get:
6,650 - .02 * b = 6,430
solve for b to get b = 220 / .02 = 11,000
since a = 35,000 - b, you get a = 24,000
since c = a + b, you get c = 35,000
you can confirm by replacing a with 24,000 and b with 11,000 and c with 35,000 in the original equations to see if they are true.
the original equations are:
a + b + c = 70,000
.09 * a + .07 * b + .10 * c = 6,430.
replace a with 24,000 and b with 11,000 and c with 35,000 to get:
24,000 + 11,000 + 35,000 = 70,000 which becomes 70,000 = 70,000 which is true.-
.09 * 24,000 + .07 * 11,000 + .10 * 35,000 = 6,430 which becomes 6,430 = 6,430 which is true.
both original equations are true, so the solution is good.
you were asked how much should be invested in bonds.
the answer is 24,000.
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