Question

If $2^a+3^b=17$ and $2^{a+2}-3^{b+1}=5,$ then find the value of a and b.

• A. $a=3andb=2.$
• B. $a=3andb=4.$
• C. $a=3andb=3.$
• D. $a=2andb=2.$

Answer 1

Answer: (A)

$a=3andb=2.$

The given equations are

$2^a+3^b=17$ .....(i)

$2^{a+2}{-3}^{b+1}=5$ ...(ii)

$⟹2^a\times 2^2-3^b\times 3=5⟹4\times 2^a-3\times 3^b=5$

We put $2^a=x$ & $3^b=y.$

Then the equations become

$x+y=17$ ...........(iii) &

$4x-3y=5$ ........(iv).

$\left(iii\right)\times 4⟶4x+4y=\mathrm{68.....}\left(v\right).\left(iv\right)-\left(v\right)⟶7y=63⟹y=9=3^2.\therefore 3^b=3^2⟹b=2.$

Again putting y=9 in (iii),

$x=17-y=17-9=8=2^3$

So ${2^a=2}^3$

i.e $a=3$.

So, $(a,b)=(3,2).$