Question 351114
Total investment is 100,000
x amount of investment is invested at 4% per year.
y amount of investment is invested at 5% per year.
z amount of investment is invested at 6% per year.


income from y and z exceeds income from x by 2,500.


first equation states that the total investment is composed of investment x, y, and z, and totals 100,000.


x + y + z = 100,000 (first equation)


second equation states that income from all of these investments equals 4,900 per year.


.04*x + .05*y + .06*z = 4,900 (second equation)


you have 2 equations in 3 unknowns.


we'll try to reduce the number of unknowns by establishing some equivalencies.


you are also given that the income from investment y and investment z exceeds the income from investment x by 2,500.


the equation to express that relationship is shown below:


.05*y + .06*z = .04*x + 2,500


we can use this equation to reduce the number of unknowns in the first 2 equations.


we'll solve for .04*x to get:


.04*x = .05*y + .06*z - 2,500


we'll solve for x to get:


x = (.05*y + .06*z - 2,500) / .04


we now have values for x and .04*x in terms of y and z.


we'll take the first equation and the second equation and substitute for x and .04*x in them as follows:


the equations are:


x + y + z = 100,000 (first equation)
.04*x + .05*y + .06*z = 4,900 (second equation)


substituting for x and .04*x in these equations gets us:


((.05*y + .06*z - 2,500) / .04) + y + z = 100,000
and:
(.05*y + .06*z - 2,500) + .05*y + .06*z = 4,900


if we solve these 2 equations in 2 unknowns simultaneously, we should be able to get our answer.


first step is to simplify these equations.


modified first equation is:


((.05*y + .06*z - 2,500) / .04) + y + z = 100,000


modified first equation simplifies to:


1.25*y + 1.5*z - 62,500 + y + z = 100,000


combine like terms to get:


2.25*y + 2.5*z - 62,500 = 100,000


add 62,500 to both sides of the equation to get:


2.25*y + 2.5*z = 162,500 (modified first equation)


modified second equation is:


(.05*y + .06*z - 2,500) + .05*y + .06*z = 4,900


modified second equation simplifies to:


.05*y + .06*z - 2,500 + .05*y + .06*z = 4,900


combine like terms to get:


.10*y + .12*z - 2,500 = 4,900


add 2,500 to both sides of the equation to get:


.10*y + .12*z = 7,400 (modified second equation).


your 2 equations that now need to be solved simultaneously are:


2.25*y + 2.5*z = 162,500 (modified first equation)
and:
.10*y + .12*z = 7,400 (modified second equation).


Multiply both sides of the second equation by 22.5 to get:


2.25*y + 2.5*z = 162,500 (modified first equation)
and:
2.25*y + 2.7*z = 166,500 (modified second equation).


Subtract the first equation from the second equation to get:


.2*z = 4,000


Divide both sides of this equation by .2 to get:


z = 4,000 / .2 = 20,000


use this value of z to solve for y in either one of the modified first equation or modified second equation (the original modified before we started multiplying them out.


those 2 original modified equations are:


2.25*y + 2.5*z = 162,500 (modified first equation)
and:
.10*y + .12*z = 7,400 (modified second equation).


In the first of these equation, let z = 20,000 to get:


2.25*y + 2.5*(20,000) = 162,500


Simplify to get:


2.25*y + 50,000 = 162,500


Subtract 50,000 from both sides of this equation to get:


2.25*y = 162,500 - 50,000


Simplify further to get:


2.25*y = 112,500


Divide both sides of this equation by 2.25 to get:


y = 50,000


If you solved for y in the second equation, you would have gotten the same answer, as you should.   I checked it out and it's good.


You now have 2 values.


y = 50,000, and z = 20,000


Since x + y + z = 100,000, that must mean that x = 30,000


Your answer should be:


x = 30,000
y = 50,000
z = 20,000


It remains only to check your answers to confirm that they are good.


Your original first equation is:


x + y + z = 100,000


That's clearly true.


Your original second equation is:


.04*x + .05*y + .06*z = 4,900


Substituting for x,y,z that we solved for, this becomes:


.04*30,000 + .05*50,000 + .06*20,000 = 4,900


This simplifies to:


1,200 + 2,500 + 1,200 = 4,900


This simplifies further to:


4,900 = 4,900 which is also true.


The third equation that we used to reduce the number of unknowns to 2 was:


.05*y + .06*z = .04*x + 2,500


Substituting for x, y, and z in this equation, we get:


.05*50,000 + .06*20,000 = .04*30,000 + 2,500


This equation simplifies to:


2,500 + 1,200 = 1,200 + 2,500


This equation simplifies to:


3,700 = 3,700 which is also true.


Our answers for x,y,z are good.


x = investment at 4% = 30,000
y = investment at 5% = 50,000
z = investment at 6% = 20,000