If 2x + 3y = 31, 2x + 2 – 3y + 1 = -65, then ……………A) x = 3

1.	If 2x + 3y = 31, 2x + 2 – 3y + 1 = -65, then ……………A) x = 3, y = 2 B) x = 3,y = 4C) x = 2, y = 3 D) x = 4, y = 3
Answer» Correct option is (C) x = 2, y = 3 Given equations are \(2^x+3^y=31\) __________(1) and \(2^{x+2}-3^{y+1}=-65\) \(\Rightarrow\) \(2^x.2^2-3^y.3=-65\) \(\Rightarrow\) \(4.2^x-3.3^y=-65\) __________(2) Take \(2^x=X\;\&\;3^y=Y\) Then equations (1) & (2) converts into X+Y = 31 __________(3) and 4X - 3Y = -65 __________(4) Put Y = 31 - X from equation (3) into equation (4), we obtain 4X - 3 (31 - X) = -65 \(\Rightarrow\) 4X - 93 + 3X = -65 \(\Rightarrow\) 7X = -65 + 93 = 28 \(\Rightarrow\) X = \(\frac{28}7\) = 4 \(\therefore\) Y = 31 - X = 31 - 4 = 27 \(\Rightarrow2^x=4\;\&\;3^y=27\) \((\because X=2^x\;\&\;Y=3^y)\) \(\Rightarrow2^x=2^2\;\&\;3^y=3^3\) \(\Rightarrow\) x = 2 & y = 3 Correct option is C) x = 2, y = 3

If 2x + 3y = 31, 2x + 2 – 3y + 1 = -65, then ……………A) x = 3, y = 2 B) x = 3,y = 4C) x = 2, y = 3 D) x = 4, y = 3

Answer»

Correct option is (C) x = 2, y = 3

Given equations are

\(2^x+3^y=31\) __________(1)

and \(2^{x+2}-3^{y+1}=-65\)

\(\Rightarrow\) \(2^x.2^2-3^y.3=-65\)

\(\Rightarrow\) \(4.2^x-3.3^y=-65\) __________(2)

Take \(2^x=X\;\&\;3^y=Y\)

Then equations (1) & (2) converts into

X+Y = 31 __________(3)

and 4X - 3Y = -65 __________(4)

Put Y = 31 - X from equation (3) into equation (4), we obtain

4X - 3 (31 - X) = -65

\(\Rightarrow\) 4X - 93 + 3X = -65

\(\Rightarrow\) 7X = -65 + 93 = 28

\(\Rightarrow\) X = \(\frac{28}7\) = 4

\(\therefore\) Y = 31 - X

= 31 - 4 = 27

\(\Rightarrow2^x=4\;\&\;3^y=27\) \((\because X=2^x\;\&\;Y=3^y)\)

\(\Rightarrow2^x=2^2\;\&\;3^y=3^3\)

\(\Rightarrow\) x = 2 & y = 3

Correct option is C) x = 2, y = 3