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.