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CAT 2019 Slot 2 — QA Question 31

IndicesEasy

If 5x - 3y = 13438 and 5x-1 + 3y+1 = 9686, then x + y equals

Answer & solution

Answer: 13

Solution

Easy

Substitute k=5x−1k=5^{x-1} and m=3ym=3^{y} so both equations become linear in k,mk,m. Note 5x=5⋅5x−1=5k5^x=5\cdot 5^{x-1}=5k and 3y+1=3⋅3y=3m3^{y+1}=3\cdot 3^{y}=3m. Solve the linear system, then read off xx and yy from the powers.

1

Linearise. Let k=5x−1k=5^{x-1} and m=3ym=3^{y}.

5x−3y=13438  ⇒  5k−m=13438(1)5x−1+3y+1=9686  ⇒  k+3m=9686(2)\begin{aligned} &5^x - 3^y = 13438 \;\Rightarrow\; 5k - m = 13438 \quad\text{(1)}\\ &5^{x-1} + 3^{y+1} = 9686 \;\Rightarrow\; k + 3m = 9686 \quad\text{(2)} \end{aligned}
2

Eliminate mm. Multiply (1) by 33 and add to (2).

15k−3m=40314[3×(1)]⇒ 16k=50000[+(2)]⇒ k=3125=55\begin{aligned} &15k - 3m = 40314 \quad\text{[}3\times\text{(1)]}\\ &\Rightarrow\ 16k = 50000 \quad\text{[}+\text{(2)]}\\ &\Rightarrow\ k = 3125 = 5^5 \end{aligned}
3

Recover xx and yy. From k=5x−1=55k=5^{x-1}=5^5 and equation (2).

5x−1=55  ⇒  x=63m=9686−3125=6561  ⇒  m=2187=373y=37  ⇒  y=7\begin{aligned} &5^{x-1}=5^5 \;\Rightarrow\; x = 6\\ &3m = 9686 - 3125 = 6561 \;\Rightarrow\; m = 2187 = 3^7\\ &3^{y}=3^7 \;\Rightarrow\; y = 7 \end{aligned}
x+y=6+7=13x+y = 6+7 = 13
CAT 2019 Slot 2 QA Q31: If 5 x - 3 y = 13438 and 5 x-1 + 3 y+1 = 9686, then x + y equals — Solution | TheCATExam