y = 8% investment.
z = 9% investment.
you have 3 equations to solve simultaneously.
they are:
x + y + z = 65000
.06*x + .08*y + .09*z = 4800
.09*z = .08*y + 600
that third equation is rearranged to become:
.08*y - .09*z = -600
your 3 equations are now:
x + y + z = 65000
.06*x + .08*y + .09*z = 4800
.08*y - .09*z = -600
put the coefficients of the variable terms and the constant terms into a row by 4 column matrix as shown below:
1 1 1 65000
.06 .08 .09 4800
0 .08 -.09 -600
use the gauss elimination method to turn this matrix into the following:
42 0 0 1,260,000
0 14 0 210,000
0 0 -21 -420,000
the first column is the coefficient of the x variable.
the second column is the coefficient of the y variable.
the third column is the coefficient of the z variable
the fourth column is the constant term.
solve for x in the first row to get x = 1,260,000 / 42 = 30,000
solve for y in the second row to get y = 210,000 / 14 = 15,000
solve for z in the third row to ge t z = -420,000 / -21 = 20,000
this was a headache to do manually, but i did it and got those results.
i also used an online gauss jordan calculator to do this mechanically and got the same results.
my manual effort was done in different steps than the mechanical effort, but the results were the same, as they should have been.
here are the results of the mechanized method.
here are the results of the manual method.
my manual effort did not go as far as the mechanical method because i didn't reduce the matrix to where all the coefficients were 1, but that didn't really matter since the same answer was gotten either way.
there are two basic methods.
gauss elimination and gauss jordan elimination.
gauss elimination is similar to what i did manually.
gauss jordan elimination goes one step further and makes all the coefficients equal to 1 so you can then just read off the answer.
my manual effort was pretty close to gauss jorden elimination with the exception that i didn't make the coefficients equal to 1.
at the point that i was at, i could have easily done that by doing the following operations.
R1 = R1 / 42
R2 = R2 / 14
R3 = R3 / -2121
the final matrix would have then been:
1 0 0 30,000
0 1 0 15,000
0 0 1 20,000
here's the link to the calculator i used for the gauss-jordan elimination method.
http://www.gregthatcher.com/Mathematics/GaussJordan.aspxhttp://www.gregthatcher.com/Mathematics/GaussJordan.aspx
note that this calculator forced me to enter fractions rather than decimals, i.e. .08 become 8/100, etc.