system of linear equations: Matrices

Question

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 corporate 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 amount in each investment. 

Answer 1

Hello, Allison.

 

Here's how to set it up:

 

Let x, y, z be the amount invested in treasury bills, treasury bonds, and corporate bonds, respectively.  Then

 

x + y + z = 10000

0.06x + 0.07y + 0.08z = 680

z = 0.5x

 

-- Jonathan

 

 

PS (10/6/14):  I wanted to post another comment or answer, but I can't seem to, so I'm editing my original answer to include what I wanted to post.

 

This system of equations can be written as a matrix equation



[ 1    1    1    ] [x]      [10000]
[ .06 .07 .08 ] [y]  =  [  680 ]
[ .5   0    -1  ] [z]       [   0   ]


which can be written as an augmented matrix


[ 1    1    1    |  10000 ]
[ .06 .07 .08  |    680  ]
[ .5   0    -1   |      0    ]

Answer 2

Let x = amount invested in treasury bills that yield 6%.

Let y = amount invested in treasury bonds that yield 7%.

Let z = amount invested in corporate bonds that yield 8%.

Carletta invests all of her $10000:

x + y + z = 10000

 

Carletta wants total yield to be $680:

0.06x + 0.07y + 0.08z = 680, or, multiplying both sides by 100,

6x + 7y + 8z = 68000

 

The amount invested in corporate bonds must be half that invested in treasury bills:

z = (1/2)x, or x = 2z

 

Let's substitute 2z for x in the other two equations:

2z + y + z = 10000

12z + 7y + 8z = 68000

Simplifying:

y + 3z = 10000

7y + 20z = 68000

Multiplying first equation by -7 and adding second equation:

-7y - 21z = -70000

7y + 20z = 68000

----------------------

0 - z = -2000

z = $2000

x = 2z = $4000

x + y + z = 10000; 4000 + y + 2000 = 10000; y = $4000

 

Check:

x + y + z =? 10000

4000 + 4000 + 2000 = 10000 √

 

0.06x + 0.07y + 0.08z =? 680

0.06(4000) + 0.07(4000) + 0.08(2000)

= 240 + 280 + 160 = 680 √

 

z =? (1/2)x

2000 =? (1/2)(4000)

(1/2)(4000) = 2000 √

Answer 3

 X = amount invested in Treasury bills. ( 6%)

 Y = amount invested in Treasury bonds ( 7%)

  Z = Corporate bounds ( 0.08)

 

   Z = X/2

    2Z + Y + Z = 10,000

   

    Y+ 3Z = 10,000

     2Z( 0.06) + ( 10,000-3Z) 0.07 + Z( 0.08)  = 680

 

    0.12Z +700- 0.21Z + Z0.08 = 680

                  - 0.01 Z = -20

 

                   Z = 2000

                    X = 4000

                    Y = 4000

 

 

Answer 4

Hey Allison -- annual income of $680 is 6.8% on the $10k ... Tbill-CB spread is 2% ...

 

a 2:1 mix means the Tbill-CB pak will yield only 1/3 thru the 2% spread => 6.67% ...

 

landing at 6.8% means going 13 basis pts thru the 33 basis pt spread to 7% Tbonds ...

 

a 20:13 mix (~60/40) works for Tbill-CB pak to Tbonds ... Tbonds 4k, Tbills 4k, CB 2k 

 

==> Check: 0.06 of 4k + 0.07 of 4k + 0.08 of 2k = 240 +280 +160 = 680 ... Regards :)