Two equations and two unknowns; solve simultaneously.
Write and balance the chemical equations.
CH4 + 2O2 ==> CO2 + H2O
C2H6 + 7O2 ==> 2CO2 + 6H2O
===========================
If we let x = pressure CH4 and y = pressure C2H6, then
x + y = 294 mm is the first math equation.
Looking at the chemical equations, if x is pressure CH4, then x = pressure of CO2 produced from the CH4 and 2y is the pressure produced by the C2H6; therefore,
x + 2y = 357 mm is th second math equation. Writing them together and solving, we have
x + y = 294
x + 2y = 357
Solve simultaneously, and I obtained
x = 231 and y = 63. You should confirm all of that.
Then since PCH4 = XCH4Ptotal
Rearrange to obtain XCH4 = (PCH4/Ptotal)
and XC2H6 = PC2H6/Ptotal.
I've estimated
XCH4 = 0.786
XC2H6 = 0.214