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If $ 999x + 888y = 1332 $
 $ 888x + 999y = 555 $ then the value of $ x + y = \_\_\_ $
A.1
B.2
C.999
D.None of these

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Last updated date: 23rd May 2024
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Answer
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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.

Complete step-by-step 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 \;
  $