A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $3,000 or $6,000. If the partnership raised $258,000, then how many investors contributed $3,000 and how many contributed $6,000?

x = $3,000 investors
y =
$6,000 investors

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 jar contains 70 nickels and dimes worth $5.70. How many of each kind of coin are in the jar?

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.

The concession stand at an ice hockey rink had receipts of $7400 from selling a total of 3000 sodas and hot dogs. If each soda sold for $2 and each hot dog sold for $3, how many of each were sold?

x= soda

y= hotdogs

1) A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $3,000 or $6,000. If the partnership raised $258,000, then how many investors contributed $3,000 and how many contributed $6,000?

x is the number of investors that contributed 3,000.

y is the number of investors that contributed 6,000.

Building the system:

There are 60 investors. So:

[tex]x + y = 60[/tex]

In all, the partnership raised $258,000. So:

[tex]3000x + 6000y = 258000[/tex]

Simplifying by 3000, we have:

[tex]x + 2y = 86[/tex]

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:

[tex]1)x + y = 60[/tex]

[tex]2)x + 2y = 86[/tex]

I am going to multiply 1) by -1, then add 1) and 2), so x is canceled.

[tex]1) - x - y = -60[/tex]

[tex]2) x + 2y = 86[/tex]

[tex]-x + x -y + 2y = -60 +86[/tex]

[tex]y = 26[/tex]

Now we get back to equation 1), and find x

[tex]x + y = 60[/tex]

[tex]x = 60-y = 60-26 = 34[/tex]

There were 34 $3,000 investors and 26 $6,000 investors.

2) A jar contains 70 nickels and dimes worth $5.70. 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 70 coins. So:

[tex]x + y = 70[/tex]

They are worth $5.70. So:

[tex]0.05x + 0.10y = 5.70[/tex]

Solving the system:

[tex]1) x+y = 70[/tex]

[tex]2) 0.05x + 0.10y = 5.70[/tex]

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

[tex]1) -0.1x -0.1y = -7[/tex]

[tex]2) 0.05x + 0.1y = 5.70[/tex]

[tex] -0.1x + 0.05x -0.1y + 0.1y = -1.3[/tex]

[tex]-0.05x = -1.3[/tex] *(-100)

[tex]5x = 130[/tex]

[tex]x = \frac{130}{5}[/tex]

[tex]x = 26[/tex]

Now:

[tex]x+y = 70[/tex]

[tex]y = 70 - x = 70 - 26 = 44[/tex]

There are 26 nickels and 44 dimes in the jar.

3) The concession stand at an ice hockey rink had receipts of $7400 from selling a total of 3000 sodas and hot dogs. If each soda sold for $2 and each hot dog sold for $3, how many of each were sold?

x is the nuber of sodas and y is the number of hot dogs.

Building the system:

3000 items were sold. So:

[tex]x + y = 3000[/tex]

$7,4000 was the total price of these items. So:

[tex]2x + 3y = 7400[/tex]

Solving the system:

[tex]1)x + y = 3000[/tex]

[tex]2)2x + 3y = 7400[/tex]

I am going to multiply 1) by -2, so we can cancel x

[tex]1) -2x -2y = -6000[/tex]

[tex]2) 2x + 3y = 7400[/tex]

[tex]-2x + 2x -2y + 3y = -6000 + 7400[/tex]

[tex]y = 1400[/tex]

Now, going back to 1)

[tex]x + y = 3000[/tex]

[tex]x = 3000-y = 3000-1400 = 1600[/tex]

1600 sodas and 1400 hot dogs were sold.