Given,
System of linear equations,
7x+11y+αz=13
5x+4y+7z=β
175x+194y+57z=361
Now we know that, for infinite solution D=D1=D2=D3=0
Now solving D=0 we get,
∣75175114194α757∣=0
Now applying operation R3→R3−25R1 we get,
∣750114−81α757−25α∣=0
⇒81(49−5α)+(57−25α)(−27)=0
⇒270α=−2430⇒α=−9
And now solving D1=0 we get,
∣13β361114194−9757∣=0
⇒13(4×57−7×194)−β(11×57+9×194)+361(11×7+9×4)=0
⇒13(−1130)−β(2373)+361(113)=0
⇒13(−10)−β(21)+361=0
⇒21β=231
⇒β=11
∴α+β+2=4