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Solve the equation without using a calculator

[tex]\displaystyle\\3^\frac{x+2}{3x-4}-2\cdot3^\frac{5x-10}{3x-4}=7[/tex]

Solve the equation without using a calculatortexdisplaystyle3fracx23x42cdot3frac5x103x47tex class=

Respuesta :

mhanifa

Answer:

  • x = 2

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Rewrite

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

Substitute

  • [tex]3^{(x + 2)/(3x - 4)} = t[/tex]

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

  • [tex]3^{(x + 2)/(3x - 4)} = 9[/tex]
  • [tex]3^{(x + 2)/(3x - 4)} = 3^2[/tex]
  • [tex](x + 2)/(3x - 4) = 2[/tex]
  • [tex]x + 2 = 6x - 8[/tex]
  • [tex]6x - x = 2 + 8[/tex]
  • [tex]5x = 10[/tex]
  • [tex]x = 2[/tex]