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Solve 2/x+1/3y=1/5; 3/x+2/3y=2 and also find ‘a’ for which y = ax – 2 - Mathematics

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Sum

Solve `\frac { 2 }{ x } + \frac { 1 }{ 3y } = \frac { 1}{ 5 }; \frac { 3 }{ x } + \frac { 2 }{ 3y } = 2` and also find ‘a’ for which y = ax – 2

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Solution

Considering 1/x = u and 1/y = v, the given system of equations becomes

`2u + \frac { v }{ 3 } = \frac { 1 }{ 5 }`

`⇒ \frac { 6u+v }{ 3 } = \frac { 1 }{ 5 }`

30u + 5v = 3 ….(1)

`3u + \frac { 2v }{ 3 } = 2 ⇒ 9u + 2v = 6 ….(2)`

Multiplying equation (1) with 2 and equation (2) with 5, we get

60u + 10v = 6 ….(3)

45u + 10v = 30 ….(4)

Subtracting equation (4) from equation (3), we get

15u = – 24

`u = -\frac { 24 }{ 15 } = -\frac { 8 }{ 5 }`

Putting `u = -\frac { 8 }{ 5 }` in equation (2), we get;

`9 × \frac { -8 }{ 5 } + 2v = 6`

`⇒ \frac { -72 }{ 5 } + 2v = 6`

`⇒ 2v = 6 + \frac { 72 }{ 5 } = \frac { 102 }{ 5 }`

`⇒ v = \frac { 51 }{ 5 }`

Here `\frac { 1 }{ x } = u = \frac { -8 }{ 5 }`

`⇒ x = \frac { -5 }{ 8 }`

And, `\frac { 1 }{ y } = v = \frac { 51 }{ 5 } ⇒ y = ⇒ \frac { 5 }{ 51 }`

Putting ` x = \frac { -5 }{ 8 } and y = \frac { 5 }{ 51 }` in y = ax – 2,

we get;

`\frac { 5 }{ 51 } = \frac { -5a }{ 8 } – 2`

`\frac { 5a }{ 8 } = – 2 – \frac { 5 }{ 51 } = \frac { -102-5 }{ 51} = \frac { -107 }{ 51 }`

`a = \frac { -107 }{ 51 } × \frac { 8 }{ 5 } = \frac { -856 }{ 255 }`

`a = \frac { -856 }{ 255 }`

Concept: Equations Reducible to a Pair of Linear Equations in Two Variables
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