Answer:
Based on the data in the table, we can set up a system of equations to represent the relationship between the number of imports (f(x)) and the number of exports (g(x)):
2x + 3y = z (Equation 1)
4x + 4y = z (Equation 2)
6x + 5y = z (Equation 3)
8x + 6y = z (Equation 4)
In this system, x represents the month number (January = 1, February = 2, etc.), y represents the number of exports, and z represents the total trade (exports plus imports). Each equation represents the data for a specific month.
To solve the system, we can use any method of solving systems of linear equations, such as substitution or elimination. However, we can also observe a pattern in the coefficients of the variables:
2 3 5 8
4 4 6 8
We can see that the coefficients of the x-term (the month number) increase by 2 each time, and the coefficients of the y-term (the number of exports) increase by 1 each time. This suggests that the equation for the nth month (where n is an integer between 1 and 4) can be expressed as:
2n + (n+2)y = z
We can test this by plugging in the values of n and y from any of the given months and verifying that the resulting value of z matches the actual value for that month. For example, if we use the data for March (n=3, y=5), we get:
2(3) + (3+2)(5) = z
6 + 25 = 31
And we can see that the actual value of z for March is indeed 31, which confirms that our equation works.
Using this equation, we can find the solution to the system for any given month by plugging in the appropriate values for n and y. For example, to find the total trade (z) for May (which would correspond to n=5), we can plug in n=5 and solve for z:
2(5) + (5+2)y = z
10 + 7y = z
To find the number of exports for May, we can plug in n=5 and solve for y:
2(5) + (5+2)y = z
10 + 7y = z
10 + 7y = 10 + 2y + 3
5y = -3
y = -3/5
However, since exports cannot be negative, this solution is not valid. This suggests that the data given in the table only applies to the months of January through April, and we cannot use this equation to find the solution for any month beyond April.
In summary, the system of equations represents the relationship between the number of imports and exports for each month from January to April. By observing the pattern in the coefficients of the equations, we can derive a general equation that can be used to find the total trade for any month between January and April. However, we cannot use this equation to find the solution for any month beyond April, as there is not enough data to determine the number of imports and exports for those months.