Question

Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y-variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question.

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?

x = $4,000 investors
y =
$8,000 investors

Solve the system by row-reducing the corresponding augmented matrix. (Enter your answers as a comma-separated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.)
leftbrace2.gif
2x + y = 17
x + y = 13

the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y-variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question.

A jar contains 80 nickels and dimes worth $6.80. How many of each kind of coin are in the jar?

x = nickels
y = dimes

Answer 1

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.

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`

In all, the partnership raised $348,000. So:

`4000x + 8000y = 348000`

I am going to simplify by 4000. So:

`x + 2y = 87`

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`

`2)x + 2y = 87`

I am going to multiply 1) by -1. So we have

`1)-x - y = -60`

`2)x + 2y = 87`

By addition, the x are going to cancel each other

`-x + x - y + 2y = -60 + 87`

`y = 27`

For x:

`x + y = 60`

`x = 60-y = 60-27 = 33`

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`

`x + y = 13`

This system has the following augmented matrix:

`\left[\begin{array}{ccc}2&1&17\\1&1&13\end{array}\right]`

To help the row reducing, i am going to swap the first with the second line:

`L1 <-> L2`

So we have:

`\left[\begin{array}{ccc}1&1&13\\2&1&17\end{array}\right]`

Now, reducing the first column.

`L2 = L2 - 2L1`

So we have:

`\left[\begin{array}{ccc}1&1&13\\0&-1&-9\end{array}\right]`

Now we do:

`L2 = -L2`

And the matrix is:

`\left[\begin{array}{ccc}1&1&13\\0&1&9\end{array}\right]`

Now to reduce the second column, we do:

`L1 = L1 - L2`

`\left[\begin{array}{ccc}1&0&4\\0&1&9\end{array}\right]`.

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`

They are worth $6.80. So:

`0.05x + 0.10y = 6.80`

Solving the system:

`1)x + y = 80`

`2)0.05x + 0.10y = 6.80`

I am going to divide 1) by -10, so we can cancel y. So:

`1)-0.10x - 0.10y = -8`

`2)0.05x + 0.10y = 6.80`

Adding:

`-0.10x + 0.05x - 0.10y + 0.10y = -8 + 6.80`

`-0.05x = -1.2` *(-100)

`5x = 120`

`x = (120)/(5)`

`x = 24`

Also

`x + y = 80`

`y = 80-x = 80-24 = 56`

There were 24 nickels and 56 dimes.