If 2x + 3y = 17; 2x + 2 – 3y + 1 = 5, then (x, y) is……………�

1.	If 2x + 3y = 17; 2x + 2 – 3y + 1 = 5, then (x, y) is………………A) (3, 2) B) (2, 3) C) (-2, 3) D) (3, -2)
Answer» Correct option is (A) (3, 2) \(2^x+3^y=17\) _________(1) and \(2^{x+2} - 3^{y+1}=5\) \(\Rightarrow\) \(2^2.2^x-3.3^y=5\) \(\Rightarrow\) \(4.2^x-3.3^y=5\) _________(2) Let \(2^x=X\;\&\;3^y=Y\) Then equations (1) & (2) converts to X + Y = 17 _________(3) and 4X - 3Y = 5 _________(4) Put Y = 17 - X from (3) into equation (4), we get 4X - 3 (17 - X) = 5 \(\Rightarrow\) 4X - 51 + 3X = 5 \(\Rightarrow\) 7X = 5+51 = 56 \(\Rightarrow\) X = \(\frac{56}7\) = 8 \(\therefore\) Y = 17 - X = 17 - 8 = 9 (From (3)) \(\therefore\) X = 8 \(\Rightarrow\) \(2^x=8=2^3\) \(\Rightarrow\) x = 3 and Y = 9 \(\Rightarrow\) \(3^y=9=3^2\) \(\Rightarrow\) y = 2 Hence, (x, y) = (3, 2) Correct option is A) (3, 2)

If 2x + 3y = 17; 2x + 2 – 3y + 1 = 5, then (x, y) is………………A) (3, 2) B) (2, 3) C) (-2, 3) D) (3, -2)

Answer»

Correct option is (A) (3, 2)

\(2^x+3^y=17\) _________(1)

and \(2^{x+2} - 3^{y+1}=5\)

\(\Rightarrow\) \(2^2.2^x-3.3^y=5\)

\(\Rightarrow\) \(4.2^x-3.3^y=5\) _________(2)

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

Then equations (1) & (2) converts to

X + Y = 17 _________(3)

and 4X - 3Y = 5 _________(4)

Put Y = 17 - X from (3) into equation (4), we get

4X - 3 (17 - X) = 5

\(\Rightarrow\) 4X - 51 + 3X = 5

\(\Rightarrow\) 7X = 5+51 = 56

\(\Rightarrow\) X = \(\frac{56}7\) = 8

\(\therefore\) Y = 17 - X

= 17 - 8 = 9 (From (3))

\(\therefore\) X = 8

\(\Rightarrow\) \(2^x=8=2^3\)

\(\Rightarrow\) x = 3

and Y = 9

\(\Rightarrow\) \(3^y=9=3^2\)

\(\Rightarrow\) y = 2

Hence, (x, y) = (3, 2)

Correct option is A) (3, 2)