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

For all the problems we are going to follow the same process that will consist on 1. Define the variables. 2. Formulate the system of two linear equations 3. Solve by the elimination method 4. Give the answer.

First problem.

- Variables:

x = number of investors that contribute with $3,000

y = number of investors that contribute with $6,000

- System of equations.

First equation will be the result of adding the number of investors that will give us a total of 60 investors then:

x + y = 60

Second equation is the result of multypling the number of investors by its invest and adding between them to get the $258,000 they have raised:

3,000x + 6,000y = 258,000

The system will be:

x + y = 60

3,000x + 6,000y = 258,000

- Solve by the elimination method.

To cancel x we are going to multiply by -3000 in both sides of the first equation and we will leave the second equation the same but in the first row to make easier the calculations then:

3,000x + 6,000y = 258,000

(x + y ) (-3,000) = 60 (-3,000) solve the multiplications

3,000x + 6,000y = 258,000

-3,000x - 3,000y = -180,000 solve the operations

0 + 3,000y = 78,000 clear y

[tex]y = \frac{78,000}{3,000}[/tex] solve the division

y = 26

For calculate x, replace the y value in the first equation and solve it

x + 26 = 60

x = 60 - 26

x = 34

- Answer

The lawyer has 34 investors that contribute with $3,000 and has 26 investors that contribute with $6,000.

Second problem

- Variables:

x = number of nickels

y = number of dimes

- System of equations.

First equation will be the result of adding the number of nickels and dimes that will give us a total of 70 coins in the jar then:

x + y = 70

Second equation is the result of multypling the number of coins by its value and adding between them to get the $5.70 that the jar worth:

0.05x + 0.1 y = 5.70

The system will be:

x + y = 70

0.05x + 0.1 y = 5.70

- Solve by the elimination method.

To cancel x we are going to multiply by -0.05 in both sides of the first equation and we will leave the second equation the same but in the first row to make easier the calculations then:

0.05x + 0.1 y = 5.70

(x + y ) (-0.05) = 70 (-0.05) solve the multiplications

0.05x + 0.1 y = 5.70

-0.05x - 0.05y = -3.50 solve the operations

0 + 0.05y = 2.20 clear y

[tex]y = \frac{2.20}{0.05}[/tex] solve the division

y = 44

For calculate x, replace the y value in the first equation and solve it

x + 44 = 70

x = 70 - 44

x = 26

- Answer

The jar has 26 nickels and 44 dimes that worth $5.70.

Third problem.

- Variables:

x = number of sodas sold by $2

y = number of hotdogs sold by $3

- System of equations.

First equation will be the result of adding the number of sodas and hotdogs that is of 3000 then:

x + y = 3000

Second equation is the result of multypling the number of sodas and hotdogs by its price and adding between them to get the $7400 of the receipts:

2x + 3y = 7400

The system will be:

x + y = 3000

2x + 3y = 7400

- Solve by the elimination method.

To cancel x we are going to multiply by -2 in both sides of the first equation and we will leave the second equation the same but in the first row to make easier the calculations then:

2x + 3y = 7400

(x + y ) (-2) = 3000 (-2) solve the multiplications

2x + 3y = 7400

-2x - 2y = -6000 solve the operations

0 + 1y = 1400 clear y

[tex]y = \frac{1400}{1}[/tex] solve the division

y = 1400

For calculate x, replace the y value in the first equation and solve it

x + 1400 = 3000

x = 3000 - 1400

x = 1600

- Answer

The concession stand has sold 1600 sodas and 1400 hotdogs