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Last updated date: 29th Nov 2023
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# If $999x + 888y = 1332$  $888x + 999y = 555$ then the value of $x + y = \_\_\_$ A.1B.2C.999D.None of these

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Hint: To find the value of $x + y$ , we need to solve the given two linear equations and find the values of x and y first. For solving these equations, we are going to use the elimination method. After we get the values of x and y, we just need to add them and we will get our answer.

In this question, we are given a system of equations with 2 linear equations and we need to find the value of $x + y$ .
Given equations are:
$999x + 888y = 1332$ - - - - - - - - - - - (1)
$888x + 999y = 555$ - - - - - - - - - - - - - (2)
We can take out 111 common in equation (1) and equation (2). Therefore, we get
$\Rightarrow 111\left( {9x + 8y} \right) = 1332$
$\Rightarrow 9x + 8y = 12$ - - - - - - - - - - (3)
And
$\Rightarrow 111\left( {8x + 9y} \right) = 555$
$\Rightarrow 8x + 9y = 5$ - - - - - - - - - - - (4)
Now, we can solve these equations using the elimination method.
For that, multiply equation (3) with 8 and equation (4) with 9, we get
$\Rightarrow \left( {9x + 8y} \right) \times 8 = 12 \times 8$
$\Rightarrow 72x + 64y = 96$ - - - - - - - - (5)
And
$\Rightarrow \left( {8x + 9y} \right) \times 9 = 5 \times 9$
$\Rightarrow 72x + 81y = 45$ - - - - - - - - (6)
Now, subtract equation (6) from equation (5), we get
$\underline 72x + 64y = 96 \\ - 72x - 81y = - 45 \\ \\ 0 - 17y = 51 \;$
$\Rightarrow - 17y = 51 \\ \Rightarrow y = - \dfrac{{51}}{{17}} \\ \Rightarrow y = - 3 \;$
Now, putting $y = - 3$ in equation (4), we get
$\Rightarrow 8x + 9\left( { - 3} \right) = 5 \\ \Rightarrow 8x - 27 = 5 \\ \Rightarrow 8x = 5 + 27 \\ \Rightarrow 8x = 32 \\ \Rightarrow x = \dfrac{{32}}{8} \\ \Rightarrow x = 4 \;$
Now, we need to find the value of $x + y$ . Therefore,
$\Rightarrow x + y = 4 + \left( { - 3} \right) \\ \Rightarrow x + y = 4 - 3 \\ \Rightarrow x + y = 1 \;$
Hence, our answer is option A.
So, the correct answer is “Option A”.

Note: We can also solve equation (3) and equation (4) using substitution method.
$\Rightarrow 9x + 8y = 12$
$\Rightarrow 8x + 9y = 5$
Now,
$\Rightarrow 9x + 8y = 12 \\ \Rightarrow 9x = 12 - 8y \\ \Rightarrow x = \dfrac{{12 - 8y}}{9} \;$
Now, putting $x = \dfrac{{12 - 8y}}{9}$ in equation (4), we get
$\Rightarrow 8x + 9y = 5 \\ \Rightarrow 8\left( {\dfrac{{12 - 8y}}{9}} \right) + 9y = 5 \\ \Rightarrow \dfrac{{96 - 64y}}{9} + 9y = 5 \\ \Rightarrow \dfrac{{96 - 64y + 81y}}{9} = 5 \\ \Rightarrow 96 + 17y = 45 \\ \Rightarrow 17y = 45 - 96 \\ \Rightarrow 17y = - 51 \\ \Rightarrow y = - 3 \;$
Now, putting $y = - 3$ in equation (4), we get
$\Rightarrow 8x + 9y = 5 \\ \Rightarrow 8x + 9\left( { - 3} \right) = 5 \\ \Rightarrow 8x - 27 = 5 \\ \Rightarrow 8x = 32 \\ \Rightarrow x = 4 \;$