Question

Solve the equation without using a calculator


`\displaystyle\\3^(x+2)/(3x-4)-2\cdot3^(5x-10)/(3x-4)=7`

Answer 1

• x = 2

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Rewrite

• (5x - 10)/(3x - 4) =
• (6x - 8 - x - 2)/(3x - 4) =
• 2 - (x + 2)/(3x - 4)

Substitute

• 
  `3^((x + 2)/(3x - 4)) = t`

New equation

• t - 2·9/t = 7
• t - 18/t = 7
• t² - 7t - 18 = 0
• t² - 9t + 2t - 18 = 0
• t(t - 9) + 2(t - 9) = 0
• (t - 9)(t + 2) = 0
• t - 9 = 0 and t + 2 = 0
• t = 9 and t = - 2 (excluded as the power of 3 can't be negative)

Substitute back

• 
  `3^((x + 2)/(3x - 4)) = 9`
• 
  `3^((x + 2)/(3x - 4)) = 3^2`
• 
  `(x + 2)/(3x - 4) = 2`
• 
  `x + 2 = 6x - 8`
• 
  `6x - x = 2 + 8`
• 
  `5x = 10`
• 
  `x = 2`