When Seven times an even number is added to an odd number, it results in $21.$ When the same even number is subtracted from thrice the odd number, $19$ is obtained. The sum of the numbers is .

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Solution

Let the odd number be $x$ and the even number be $y$ .
Seven times the even number when added to the odd number results $21.$
$7 y + x = 21 . . . (a)$

When the even number is subtracted from thrice the odd number, $19$ is obtained.
$3 x - y = 19 . . . (b)$

From (a) we get,
$x = 21 - 7 y$

Substituting $x$ in (b) we get,
$3 (21 - 7 y) - y = 19$
$\Rightarrow 63 - 21 y - y = 19$
$\Rightarrow - 22 y = 19 - 63$
$\Rightarrow - 22 y = - 44$
$\Rightarrow y = 2$

Substituting $y$ in (a) we get,
$x = 21 - 7 (2) = 21 - 14 = 7$

The addition of the odd and even numbers is $7 + 2 = 9.$

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