let x and y be the lengths of the side of the rectangular garden and x>y

the perimeter of a garden is 68 m

2(x + y) = 68 divide both sides by 2

x + y = 34

the longer side is 10 times greater than the smaller side

x = 10y

by solving the system of equations

x + y = 34

x = 10y

we find

x = 30.9 m

y = 3.09 m

click here to see the step by step solution of the system of equations

the measure of the sides are 30.9 m and 3.09 m.