Solve the following system of equations by using the method of elimination by equating the co-efficients. (x/y)+(2y/5)+2=10;(2x/7)–(5/2)+1=9 - Mathematics

Sum

Solve the following system of equations by using the method of elimination by equating the co-efficients.

\frac { x }{ y } + \frac { 2y }{ 5 } + 2 = 10; \frac { 2x }{ 7 } – \frac { 5 }{ 2 } + 1 = 9

Solution

The given system of equation is

\frac { x }{ y } + \frac { 2y }{ 5 } + 2 = 10 ⇒ \frac { x }{ y } + \frac { 2y }{ 5 } = 8 …(1)

\frac { 2x }{ 7 } – \frac { 5 }{ 2 } + 1 = 9 ⇒ \frac { 2x }{ 7 } –\frac { 5 }{ 2 } = 8 ….(2)

The equation (1) can be expressed as :

\frac { 5x+4y }{ 10 } = 8 ⇒ 5x + 4y = 80 ….(3)

Similarly, the equation (2) can be expressed as :

\frac { 4x-7y }{ 14 } = 8 ⇒ 4x – 7y = 112 ….(4)

Now the new system of equations is

5x + 4y = 80 ….(5)

4x – 7y = 112 ….(6)

Now multiplying equation (5) by 4 and equation (6) by 5, we get

20x – 16y = 320 ….(7)

20x + 35y = 560 ….(8)

Subtracting equation (7) from (8), we get ;

y = -\frac { 240 }{ 51 }

Putting y = -\frac { 240 }{ 51 }  in equation (5), we get ;

5x + 4 × \frac { -240 }{ 51 } = 80 ⇒ 5x – \frac { 960 }{ 51 } = 80

⇒ 5x = 80 + \frac { 960 }{ 51 } = \frac { 4080+960 }{ 51 } = \frac {5040 }{ 51 }

⇒ x = \frac { 5040 }{ 255 } = \frac { 1008 }{ 51 }= \frac { 336 }{ 17 }

⇒ x = \frac { 336 }{ 17 }

Hence, the solution of the system of equations is, x = \frac { 336 }{ 17}, y = -\frac { 80 }{ 17 }

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Elimination Method
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