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Let's set up the system of linear equations based on the information given:
3x + 4y = 48
6x + 2y = 60
Now, let's solve this system of equations. First, let's multiply the first equation by 2 to make the coefficients of y in both equations the same:
6x + 8y = 96
6x + 2y = 60
Now, subtract the second equation from the first:
6x + 8y - 6x - 2y = 96 - 60
6y = 36
y = 6
Now that we have the value of y, let's substitute it back into the first equation to solve for x:
3x + 4(6) = 48
3x + 24 = 48
3x = 24
x = 8
Therefore, a burger meal costs $8 and a hot dog meal costs $6.