Answer:
1) There were 33 $4,000 investors and 27 $8,000 investors.
2) The solution in x = 4, y = 9.
3) There were 24 nickels and 56 dimes.
Step-by-step explanation:
1) A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $4,000 or $8,000. If the partnership raised $348,000, then how many investors contributed $4,000 and how many contributed $8,000?
I am going to say that:
x is the number of investors that contributed 4,000.
y is the number of investors that contributed 8,000.
Building the system:
There are 60 investors. So:
![x + y = 60](https://tex.z-dn.net/?f=x%20%2B%20y%20%3D%2060)
In all, the partnership raised $348,000. So:
![4000x + 8000y = 348000](https://tex.z-dn.net/?f=4000x%20%2B%208000y%20%3D%20348000)
I am going to simplify by 4000. So:
![x + 2y = 87](https://tex.z-dn.net/?f=x%20%2B%202y%20%3D%2087)
Solving the system:
The elimination method is a method in which we can transform the system such that one variable can be canceled by addition. So:
![1)x + y = 60](https://tex.z-dn.net/?f=1%29x%20%2B%20y%20%3D%2060)
![2)x + 2y = 87](https://tex.z-dn.net/?f=2%29x%20%2B%202y%20%3D%2087)
I am going to multiply 1) by -1. So we have
![1)-x - y = -60](https://tex.z-dn.net/?f=1%29-x%20-%20y%20%3D%20-60)
![2)x + 2y = 87](https://tex.z-dn.net/?f=2%29x%20%2B%202y%20%3D%2087)
By addition, the x are going to cancel each other
![-x + x - y + 2y = -60 + 87](https://tex.z-dn.net/?f=-x%20%2B%20x%20-%20y%20%2B%202y%20%3D%20-60%20%2B%2087)
![y = 27](https://tex.z-dn.net/?f=y%20%3D%2027)
For x:
![x + y = 60](https://tex.z-dn.net/?f=x%20%2B%20y%20%3D%2060)
![x = 60-y = 60-27 = 33](https://tex.z-dn.net/?f=x%20%3D%2060-y%20%3D%2060-27%20%3D%2033)
There were 33 $4,000 investors and 27 $8,000 investors.
2) Solve the system by row-reducing the corresponding augmented matrix.
![2x + y = 17](https://tex.z-dn.net/?f=2x%20%2B%20y%20%3D%2017)
![x + y = 13](https://tex.z-dn.net/?f=x%20%2B%20y%20%3D%2013)
This system has the following augmented matrix:
![\left[\begin{array}{ccc}2&1&17\\1&1&13\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%261%2617%5C%5C1%261%2613%5Cend%7Barray%7D%5Cright%5D)
To help the row reducing, i am going to swap the first with the second line:
![L1 L2](https://tex.z-dn.net/?f=L1%20%3C-%3E%20L2)
So we have:
![\left[\begin{array}{ccc}1&1&13\\2&1&17\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%261%2613%5C%5C2%261%2617%5Cend%7Barray%7D%5Cright%5D)
Now, reducing the first column.
![L2 = L2 - 2L1](https://tex.z-dn.net/?f=L2%20%3D%20L2%20-%202L1)
So we have:
![\left[\begin{array}{ccc}1&1&13\\0&-1&-9\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%261%2613%5C%5C0%26-1%26-9%5Cend%7Barray%7D%5Cright%5D)
Now we do:
![L2 = -L2](https://tex.z-dn.net/?f=L2%20%3D%20-L2)
And the matrix is:
![\left[\begin{array}{ccc}1&1&13\\0&1&9\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%261%2613%5C%5C0%261%269%5Cend%7Barray%7D%5Cright%5D)
Now to reduce the second column, we do:
![L1 = L1 - L2](https://tex.z-dn.net/?f=L1%20%3D%20L1%20-%20L2)
.
So the solution is:
x = 4, y = 9.
3) A jar contains 80 nickels and dimes worth $6.80. How many of each kind of coin are in the jar?
I am going to say that x is the number of nickels and y is the number of dimes.
Each nickel is worth 5 cents and each dime is worth 10 cents.
Building the system:
There are 80 coins in all:
![x + y = 80](https://tex.z-dn.net/?f=x%20%2B%20y%20%3D%2080)
They are worth $6.80. So:
![0.05x + 0.10y = 6.80](https://tex.z-dn.net/?f=0.05x%20%2B%200.10y%20%3D%206.80)
Solving the system:
![1)x + y = 80](https://tex.z-dn.net/?f=1%29x%20%2B%20y%20%3D%2080)
![2)0.05x + 0.10y = 6.80](https://tex.z-dn.net/?f=2%290.05x%20%2B%200.10y%20%3D%206.80)
I am going to divide 1) by -10, so we can cancel y. So:
![1)-0.10x - 0.10y = -8](https://tex.z-dn.net/?f=1%29-0.10x%20-%200.10y%20%3D%20-8)
![2)0.05x + 0.10y = 6.80](https://tex.z-dn.net/?f=2%290.05x%20%2B%200.10y%20%3D%206.80)
Adding:
![-0.10x + 0.05x - 0.10y + 0.10y = -8 + 6.80](https://tex.z-dn.net/?f=-0.10x%20%2B%200.05x%20-%200.10y%20%2B%200.10y%20%3D%20-8%20%2B%206.80)
*(-100)
![5x = 120](https://tex.z-dn.net/?f=5x%20%3D%20120)
![x = \frac{120}{5}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B120%7D%7B5%7D)
![x = 24](https://tex.z-dn.net/?f=x%20%3D%2024)
Also
![x + y = 80](https://tex.z-dn.net/?f=x%20%2B%20y%20%3D%2080)
![y = 80-x = 80-24 = 56](https://tex.z-dn.net/?f=y%20%3D%2080-x%20%3D%2080-24%20%3D%2056)
There were 24 nickels and 56 dimes.