Question

Given that \[x\] and \[y\] satisfy the relation.
\[y = 3\left[ x \right] + 7\]
\[y = 4\left[ {x - 3} \right] + 4\] then find \[\left[ {x + y} \right]\].

Answer 1

- Hint: In this problem, we need to use the property of the greatest integers to solve the given equation and find the value of \[x\,\,{\text{and}}\,\,y\]. Now, again use the property of the greatest integer function to obtain the value of the given expression.

Complete step-by-step solution -
The given equations are shown below.
\[
  y = 3\left[ x \right] + 7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( 1 \right) \\
  y = 4\left[ {x - 3} \right] + 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( 2 \right) \\
\]
Now, from equation (1) and (2),
\[
  \,\,\,\,\,\,3\left[ x \right] + 7 = 4\left[ {x - 3} \right] + 4 \\
   \Rightarrow 3\left[ x \right] + 7 = 4\left[ x \right] - 12 + 4 \\
   \Rightarrow 3\left[ x \right] + 7 = 4\left[ x \right] - 8 \\
   \Rightarrow 4\left[ x \right] - 3\left[ x \right] = 8 + 7 \\
   \Rightarrow \left[ x \right] = 15 \\
   \Rightarrow 15 < x < 16 \\
\]
Now, substitute, 15 for \[\left[ x \right]\] in equation (1) to obtain the value of y.
\[
  \,\,\,\,\,\,\,y = 3\left( {15} \right) + 7 \\
   \Rightarrow y = 45 + 7 \\
   \Rightarrow y = 52 \\
\]
Substitute 15 for \[\left[ x \right]\] and 52 for y in \[\left[ {x + y} \right]\].
\[
  \,\,\,\,\,\,\,\left[ {15 + 52} \right] \\
   \Rightarrow \left[ {67} \right] \\
   \Rightarrow 67 \\
\]

Thus, the value of \[\left[ {x + y} \right]\] is 67.

Note: The greatest integer function rounds of the real numbers down to the integer less than the original number. For the interval \[\left( {n,n + 1} \right)\], the value of the greatest integer function is\[n\], where \[n\] is an integer.