What is a counterexample? "If a number is an integer, then it is either positive or negative."
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Answer:
1. The lawyer has 34 investors that contribute with $3,000 and has 26 investors that contribute with $6,000.
2. The jar has 26 nickels and 44 dimes that worth $5.70.
3. The concession stand has sold 1600 sodas and 1400 hotdogs
Step-by-step explanation:
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) <em> solve the multiplications </em>
3,000x + 6,000y = 258,000
<u>-3,000x - 3,000y = -180,000 </u><em> solve the operations</em>
0 + 3,000y = 78,000 <em>clear y</em>
<em>solve the division</em>
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) <em> solve the multiplications </em>
0.05x + 0.1 y = 5.70
<u>-0.05x - 0.05y = -3.50 </u><em> solve the operations</em>
0 + 0.05y = 2.20 <em>clear y</em>
<em>solve the division</em>
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) <em> solve the multiplications </em>
2x + 3y = 7400
<u>-2x - 2y = -6000 </u><em> solve the operations</em>
0 + 1y = 1400 <em>clear y</em>
<em>solve the division</em>
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