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16 Sep 2014, 00:14
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Shin visits an upscale fast-food restaurant where he can purchase a combination of burgers, shakes, and colas. He notices that a meal consisting of 3 burgers, 7 shakes, and one cola costs $120, while a meal with 4 burgers, 10 shakes, and one cola costs $164.50. What is the cost of a single meal that includes one burger, one shake, and one cola at this restaurant?
A. $21
B. $27
C. $31
D. $41
E. It cannot be determined
A. $21
B. $27
C. $31
D. $41
E. It cannot be determined
03 Sep 2018, 11:19
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buan15
There's a mechanical way of solving any question like this without relying on hit and trial based on linear algebra
The idea is that we are manipulating the equations by multiplying with a constant, and then adding the 2 equations to get our desired equation.
Our desired equation is one that has the same coefficient for B, S and C
eq 1 - 3B + 7S + C multiply this equation by x
eq 2 - 4B + 10S + C multiply this equation by y
Since we want the coefficients the same after adding, 7x+ 4y (coefficient of B) = 7x + 10y (coefficient of S) = x+y (coefficient of C)
this gives x = -3y/2
Since for simplicity we want x & y to be integers, we can use y = -2, so x = 3
So if we multiply eq 1 by 3 and eq 2 by -2, and add the two equations together we will get B + S + C.
Not sure if my explanation is complicated but in my opinion its a very easy and foolproof way to solve these type of questions.
16 Sep 2014, 00:14
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Official Solution:
Shin visits an upscale fast-food restaurant where he can purchase a combination of burgers, shakes, and colas. He notices that a meal consisting of 3 burgers, 7 shakes, and one cola costs $120, while a meal with 4 burgers, 10 shakes, and one cola costs $164.50. What is the cost of a single meal that includes one burger, one shake, and one cola at this restaurant?
A. $21
B. $27
C. $31
D. $41
E. It cannot be determined
Let's assume the price of a burger, shake, and cola to be \(B\), \(S\), and \(C\), respectively. We are asked to find the value of \(B+S+C=31\) given the following two equations:
\(3B+7S+C = 120\)
\(4B+10S+C = 164.5\)
We can eliminate \(C\) by subtracting the first equation from the second, resulting in \(B+3S=44.5\). Now we can subtract this equation twice from the first equation or three times from the second equation to obtain \(B+S+C=31\). This means we can determine the cost of a meal that includes one burger, one shake, and one cola without finding the individual price of each item. Therefore, the cost of such a meal is $31.
Answer: C
Shin visits an upscale fast-food restaurant where he can purchase a combination of burgers, shakes, and colas. He notices that a meal consisting of 3 burgers, 7 shakes, and one cola costs $120, while a meal with 4 burgers, 10 shakes, and one cola costs $164.50. What is the cost of a single meal that includes one burger, one shake, and one cola at this restaurant?
A. $21
B. $27
C. $31
D. $41
E. It cannot be determined
Let's assume the price of a burger, shake, and cola to be \(B\), \(S\), and \(C\), respectively. We are asked to find the value of \(B+S+C=31\) given the following two equations:
\(3B+7S+C = 120\)
\(4B+10S+C = 164.5\)
We can eliminate \(C\) by subtracting the first equation from the second, resulting in \(B+3S=44.5\). Now we can subtract this equation twice from the first equation or three times from the second equation to obtain \(B+S+C=31\). This means we can determine the cost of a meal that includes one burger, one shake, and one cola without finding the individual price of each item. Therefore, the cost of such a meal is $31.
Answer: C
