You can solve this problem by creating 2 equations with 2 unknowns and solving the system you create.
Let use the variable w to represent the width of the original rectangle and the variable y to represent the length of the original rectangle.
You can write one equation from the information that says the perimeter is equal to 10. Since perimeter is equal to 2 times the length plus 2 times the width, we can write:
2y + 2w =10
Then using the information that says the length is halved and width doubled for a perimeter of 4 more (or 14), we can write:
y/2 + y/2 + 2w + 2w = 14
We can combine like terms in this equation as follows:
y + 4w = 14
Now we have two equations with two unknowns and can use elimination to solve:
2y + 2w =10
y + 4w = 14
Multiply the second equation by -2 and solve for w:
2y + 2w =10
-2y - 8w = -28
-6w = -18
w = 3
Since 2y + 2w = 10 for the original rectangle, we can solve for y (the length):
2y + 2w = 10
2y + 2*3 =10
2y + 6 = 10
2y + 6 - 6 =10 - 6
2y = 4
y = 2
The length of the original rectangle will be 2 and the width will be 3.
NOTE: The "names" of these sides is not in line with the length being longer than the width.
