Show step by step 1 log6x4log6x216log10100 2 Solve the follo

Show step by step

1. log6(x-4)-log6(x^2-16)=log10(100)

2. Solve the following system of equations

-y+x=3

(x-2)^2+(y+3)^2=4

1) log₆(x-4)-log₆(x²-16) = log₁₀(100)

log₆(x-4)/(x²-16) = log₁₀10² [ log_xa- log_xb = log_x(a/b) ]

log₆[(x-4)/(x+4)(x-4)] = 2 log₁₀10

log₆[1/(x+4)] = 2

1/ (x+4) = 6²

x+ 4 = 1/36

x= (1/36)-4 = 1-144/36 = -143/36

x = -143/36

2 ) Given linear system of equations are

-y+x=3 -----(1)

i.e x = 3 + y

(x-2)²+(y+3)² = 4 ----(2)

x² - 2.x.2 + 2² + y² + 2.y.3 +3² = 4

x² - 4x + 4+ y² + 6y + 9 = 4

x² - 4x + y² + 6y + 4 + 9 - 4 = 0

x² - 4x + y² + 6y + 9 = 0

put x = y + 3 in this equation

( y + 3 )2 - 4 . ( y + 3 ) + y² + 6y + 9 = 0

y² + 9 + 6y - 4y - 12 + y² + 6y + 9 = 0

2y² + 8y + 6 = 0

y² + 4y + 3 = 0

( y + 3 ) ( y + 1 ) = 0

y = -3 ,-1

put y = -3 in x = 3 + y

x = 3 - 3 = 0

put y = -1 in x = 3 + y

x = 3 - 1 = 2

Therefore,

x = 0 , 2

y = -3 , -1