Answer:
x = 5 Or x = \frac{5}{2}
2
5
Explanation:
Given \frac{x-1}{x-2}+\frac{x-3}{x-4}=\frac{10}{3}
x−2
x−1
+
x−4
x−3
=
3
10
\implies \frac{(x-1)(x-4)+(x-3)(x-2)}{(x-2)(x-4)}=\frac{10}{3}⟹
(x−2)(x−4)
(x−1)(x−4)+(x−3)(x−2)
=
3
10
\implies \frac{x^{2}-4x-x+4+x^{2}-2x-3x+6}{x^{2}-4x-2x+8}=\frac{10}{3}⟹
x
2
−4x−2x+8
x
2
−4x−x+4+x
2
−2x−3x+6
=
3
10
\implies \frac{2x^{2}-10x+10}{x^{2}-6x+8}=\frac{10}{3}⟹
x
2
−6x+8
2x
2
−10x+10
=
3
10
\implies 3(2x^{2}-10x+10)=10(x^{2}-6x+8)⟹3(2x
2
−10x+10)=10(x
2
−6x+8)
\implies 6x^{2}-30x+30=10x^{2}-60x+80⟹6x
2
−30x+30=10x
2
−60x+80
\implies 0= 10x^{2}-60x+80-6x^{2}+30x-30⟹0=10x
2
−60x+80−6x
2
+30x−30
\implies 4x^{2}-30x+50=0⟹4x
2
−30x+50=0
Divide each term by 2 ,we get
\implies 2x^{2}-15x+25=0⟹2x
2
−15x+25=0
Splitting the middle term,we get
\implies 2x^{2}-10x-5x+25=0⟹2x
2
−10x−5x+25=0
\implies 2x(x-5)-5(x-5)=0⟹2x(x−5)−5(x−5)=0
\implies (x-5)(2x-5)=0⟹(x−5)(2x−5)=0
\implies x-5 = 0 \: Or \: 2x-5 = 0⟹x−5=0Or2x−5=0
\implies x = 5 \: Or \: x=\frac{5}{2}⟹x=5Orx=
2
5
Therefore,
x = 5 Or x = \frac{5}{2}
2
5
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