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Answers
Question:
Solve the given equation:
\displaystyle{\sf\:\dfrac{\dfrac{x}{3}\:-\:\dfrac{2}{5}}{\dfrac{3}{4}\:-\:2x}\:=\:\dfrac{16}{15}}
4
3
−2x
3
x
−
5
2
=
15
16
Answer:
The value of x is
\displaystyle{\implies\boxed{\red{\sf\:x\:=\:\dfrac{18}{37}}}}⟹
x=
37
18
Step-by-step-explanation:
The given equation is \displaystyle\sf\:\dfrac{\dfrac{x}{3}\:-\:\dfrac{2}{5}}{\dfrac{3}{4}\:-\:2x}\:=\:\dfrac{16}{15}
4
3
−2x
3
x
−
5
2
=
15
16
.
Now,
\displaystyle{\sf\:\dfrac{\dfrac{x}{3}\:-\:\dfrac{2}{5}}{\dfrac{3}{4}\:-\:2x}\:=\:\dfrac{16}{15}}
4
3
−2x
3
x
−
5
2
=
15
16
\displaystyle{\implies\sf\:\dfrac{\dfrac{5x\:-\:6}{15}}{\dfrac{3\:-\:8x}{4}}\:=\:\dfrac{16}{15}}⟹
4
3−8x
15
5x−6
=
15
16
\displaystyle{\implies\sf\:\dfrac{5x\:-\:6}{15}\:\times\:\dfrac{4}{3\:-\:8x}\:=\:\dfrac{16}{15}}⟹
15
5x−6
×
3−8x
4
=
15
16
\displaystyle{\implies\sf\:\dfrac{4\:\times\:(\:5x\:-\:6\:)}{15\:\times\:(\:3\:-\:8x\:)}\:=\:\dfrac{16}{15}}⟹
15×(3−8x)
4×(5x−6)
=
15
16
\displaystyle{\implies\sf\:\dfrac{20x\:-\:24}{45\:-\:120x}\:=\:\dfrac{16}{15}}⟹
45−120x
20x−24
=
15
16
\displaystyle{\implies\sf\:15\:\times\:(\:20x\:-\:24\:)\:=\:16\:\times\:(\:45\:-\:120x\:)}⟹15×(20x−24)=16×(45−120x)
\displaystyle{\implies\sf\:300x\:-\:360\:=\:720\:-\:1920x}⟹300x−360=720−1920x
\displaystyle{\implies\sf\:300x\:+\:1920x\:=\:720\:+\:360}⟹300x+1920x=720+360
\displaystyle{\implies\sf\:2220x\:=\:1080}⟹2220x=1080
\displaystyle{\implies\sf\:x\:=\:{\dfrac{108\cancel{0}}{222\cancel{0}}}}⟹x=
222
0
108
0
\displaystyle{\implies\sf\:x\:=\:\cancel{\dfrac{108}{222}}}⟹x=
222
108
\displaystyle{\implies\boxed{\red{\sf\:x\:=\:\dfrac{18}{37}}}}⟹
x=
37
18
─────────────────────
Verification:
The given equation is
\displaystyle{\sf\:\dfrac{\dfrac{x}{3}\:-\:\dfrac{2}{5}}{\dfrac{3}{4}\:-\:2x}\:=\:\dfrac{16}{15}}
4
3
−2x
3
x
−
5
2
=
15
16
The value of x is \displaystyle\sf\:\dfrac{18}{37}
37
18
By substituting this value of x in the LHS of the given equation, we get,
\displaystyle\sf\:LHS\:=\:\dfrac{\dfrac{x}{3}\:-\:\dfrac{2}{5}}{\dfrac{3}{4}\:-\:2x}LHS=
4
3
−2x
3
x
−
5
2
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{\dfrac{\dfrac{18}{37}}{3}\:-\:\dfrac{2}{5}}{\dfrac{3}{4}\:-\:2\:\times\:\dfrac{18}{37}}}⟹LHS=
4
3
−2×
37
18
3
37
18
−
5
2
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{\dfrac{\cancel{18}}{37}\:\times\:\dfrac{1}{\cancel{3}}\:-\:\dfrac{2}{5}}{\dfrac{3}{4}\:-\:\dfrac{39}{37}}}⟹LHS=
4
3
−
37
39
37
18
×
3
1
−
5
2
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{\dfrac{6}{37}\:-\:\dfrac{2}{5}}{\dfrac{3}{4}\:-\:\dfrac{36}{37}}}⟹LHS=
4
3
−
37
36
37
6
−
5
2
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{\dfrac{6\:\times\:5\:-\:2\:\times\:37}{37\:\times\:5}}{\dfrac{3\:\times\:37\:-\:36\:\times\:4}{4\:\times\:37}}}⟹LHS=
4×37
3×37−36×4
37×5
6×5−2×37
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{\dfrac{30\:-\:74}{185}}{\dfrac{111\:-\:144}{148}}}⟹LHS=
148
111−144
185
30−74
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{\dfrac{-\:44}{185}}{\dfrac{-\:33}{148}}}⟹LHS=
148
−33
185
−44
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{\cancel{-}\:44}{185}\:\times\:\dfrac{148}{\cancel{-}\:33}}⟹LHS=
185
−
44
×
−
33
148
\displaystyle{\implies\sf\:LHS\:=\:\cancel{\dfrac{\cancel{44}}{185}}\:\times\:\dfrac{148}{\cancel{33}}}⟹LHS=
185
44
×
33
148
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{4}{185}\:\times\:\dfrac{148}{3}}⟹LHS=
185
4
×
3
148
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{4\:\times\:148}{185\:\times\:3}}⟹LHS=
185×3
4×148
\displaystyle{\implies\sf\:LHS\:=\:\cancel{\dfrac{592}{555}}}⟹LHS=
555
592
\displaystyle{\implies\sf\:LHS\:=\:\dfrac{16}{15}}⟹LHS=
15
16
\displaystyle{\implies\sf\:RHS\:=\:\dfrac{16}{15}}⟹RHS=
15
16
\displaystyle{\implies\boxed{\red{\sf\:LHS\:=\:RHS}}}⟹
LHS=RHS
Hence verified!