Question

Find the value of k for which a straight line 2x-ky=3 passes through the intersection of 2 curves 3^x+y=81 and 81^x-y=3.

Answer 1

3^(x+y) =81
3^(x+y)=3^4

so x+y=4 is one equation

81^(x-y) = 3
(3^4)^(x-y)=3

soequating the powers

4(x-y)=1

x-y = 1/4 is the second equation.

Find values of x and y

Add two equation,
X+y+x-y = 4+(1/4)
2x=17/4
x=17/8

so y= 4 - (17/8)
=(32/8) -(17/8) = 15/8

so point (17/8 , 15/8)
is the intersection of those lines.

Like 2x-ky=3 is passing through thus point .

so it can be written as

2(17/8)-k(15/8)=3

34/8 - 15k/8 =24/8
15k/8 = 34/8 - 24/8=10/8

so 15k = 10
k=10 / 15 =2/3

is the answer.

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