solve the following pairs of equations by reducing them to a pair of linear eqyatuons

Answers
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Question:
Solve the following pairs of equations by reducing them to a pair of linear equations.
( 1 / 2x ) + ( 1 / 3y ) = 2
( 1 / 3x ) + ( 1 / 2y ) = 13 / 6
Answer:
The solution of the given equations is
( x, y ) = ( 1 / 2, 1 / 3 ).
Step-by-step-explanation:
The given linear equations are
( 1 / 2x ) + ( 1 / 3y ) = 2 - - - ( 1 ) &
( 1 / 3x ) + ( 1 / 2y ) = 13 / 6 - - - ( 2 )
By substituting 1 / x = a and 1 / y = b, equation ( 1 ) becomes
( 1 / 2 ) a + ( 1 / 3 ) b = 2
⇒ ( a / 2 ) + ( b / 3 ) = 2
⇒ ( 3a + 2b ) / ( 2 * 3 ) = 2
⇒ ( 3a + 2b ) / 6 = 2
⇒ 3a + 2b = 2 * 6
⇒ 3a + 2b = 12
⇒ 2b = 12 - 3a
⇒ b = ( 12 - 3a ) / 2 - - - ( 3 )
By substituting 1 / x = a and 1 / y = b, equation ( 2 ) becomes
( 1 / 3 ) a + ( 1 / 2 ) b = 13 / 6
⇒ ( a / 3 ) + ( b / 2 ) = 13 / 6
⇒ ( 2a + 3b ) / ( 2 * 3 ) = 13 / 6
⇒ ( 2a + 3b ) / 6 = 13 / 6
⇒ 2a + 3b = 13 / 6 * 6
⇒ 2a + 3b = 13
⇒ 2a + 3 ( 12 - 3a ) / 2 = 13 - - - [ From ( 3 ) ]
⇒ 2a + ( 36 - 9a ) / 2 = 13
⇒ ( 4a + 36 - 9a ) / 2 = 13
⇒ 4a - 9a + 36 = 13 * 2
⇒ - 5a + 36 = 26
⇒ 36 - 26 = 5a
⇒ 10 = 5a
⇒ a = 10 / 5
⇒ a = 2
By substituting a = 2 in equation ( 3 ),
b = ( 12 - 3a ) / 2 - - - ( 3 )
⇒ b = ( 12 - 3 * 2 ) / 2
⇒ b = ( 12 - 6 ) / 2
⇒ b = 6 / 2
⇒ b = 3
Now,
1 / x = a
⇒ 1 / a = x
⇒ x = 1 / 2
And,
1 / y = b
⇒ 1 / b = y
⇒ y = 1 / 3
∴ The solution of the given equations is
( x, y ) = ( 1 / 2, 1 / 3 ).