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Mathematics

Solve the following equation and check your answer:

23(3x−2)=45(2x−3)−43\dfrac{2}{3}(3x - 2) = \dfrac{4}{5}(2x - 3)-\dfrac{4}{3}

Linear Eqns One Variable

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Answer

We have:

=23(3x−2)=45(2x−3)−43⇒6x−43=12(2x−3)−2015⇒6x−43=24x−36−2015⇒6x−43=24x−5615⇒15(6x−4)=3(24x−56)[By cross multiplication]⇒90x−60=72x−168⇒90x−72x=60−168[Transposing -60 to RHS and +72x to LHS]⇒18x=−108⇒x=−10818\phantom{=} \dfrac{2}{3}(3x - 2) = \dfrac{4}{5}(2x - 3)-\dfrac{4}{3} \\[1em] \Rightarrow \dfrac{6x - 4}{3} = \dfrac{12(2x - 3) - 20}{15} \\[1em] \Rightarrow \dfrac{6x - 4}{3} = \dfrac{24x - 36 - 20}{15} \\[1em] \Rightarrow \dfrac{6x - 4}{3} = \dfrac{24x - 56}{15} \\[1em] \Rightarrow 15(6x - 4) = 3(24x - 56) \quad \text{[By cross multiplication]} \\[1em] \Rightarrow 90x - 60 = 72x - 168 \\[1em] \Rightarrow 90x - 72x = 60 - 168 \quad \text{[Transposing -60 to RHS and +72x to LHS]} \\[1em] \Rightarrow 18x = -108 \\[1em] \Rightarrow x = \dfrac{-108}{18} \\[1em]

∴ x = -6

Check:

LHS=23(3x−2)LHS=23(−20)LHS=−403RHS=45(2x−3)−43RHS=45(−15)−43RHS=−12−43RHS=−403\text{LHS} = \dfrac{2}{3}(3x - 2) \\[1em] \phantom{\text{LHS}} = \dfrac{2}{3}(-20) \\[1em] \phantom{\text{LHS}} = -\dfrac{40}{3} \\[2em] \text{RHS} = \dfrac{4}{5}(2x - 3) - \dfrac{4}{3} \\[1em] \phantom{\text{RHS}} = \dfrac{4}{5}(-15) - \dfrac{4}{3} \\[1em] \phantom{\text{RHS}} = -12 - \dfrac{4}{3} \\[1em] \phantom{\text{RHS}} = -\dfrac{40}{3}

Hence, LHS = RHS.

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