The Correct Answer and Explanation is:
The correct answer is 4.
This problem provides a system of two linear equations and asks for the value of the sum x + y. The given equations are:
- 7x + 8y = 25
- 8x + 7y = 35
While you could solve for the individual values of x and y using methods like substitution or elimination, there is a more direct and efficient strategy for this specific type of problem. The key is to observe the structure of the equations. Notice that the coefficients of the x and y variables are swapped between the two equations. In the first equation, the coefficient of x is 7 and y is 8; in the second, the coefficient of x is 8 and y is 7.
This symmetrical pattern is a strong hint that adding the two equations together will simplify the problem significantly. Let’s perform this addition:
(7x + 8y) + (8x + 7y) = 25 + 35
Next, we can group the like terms on the left side of the equation:
(7x + 8x) + (8y + 7y) = 60
Combining these terms gives us a much simpler equation:
15x + 15y = 60
Now, you can see that both terms on the left side share a common factor of 15. We can factor this common coefficient out:
15(x + y) = 60
The problem asks for the value of x + y, which is exactly the expression we have inside the parentheses. To isolate this expression, we simply need to divide both sides of the equation by 15:
x + y = 60 / 15
x + y = 4
This method allows us to find the value of x + y without ever needing to calculate x and y individually, saving time and reducing the chance of calculation errors. Therefore, the value of x + y is 4, which corresponds to option (D).
