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# M is a set of six consecutive even integers. When the least three integers of set $M$ are summed, the result is $x$. When the greatest three integers of set $M$ are summed, the result is $y$. Mark the true equation.A.$y=x-18$B.$y=x+18$C.$y=2x$D.$y=2x+4$

Last updated date: 13th Jun 2024
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Answer
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Hint: We denote the six even consecutive integers from the set $M$ in ascending order as$n,n+2,n+4,n+6,n+8,n+10$. We add the first three integers and equate to $x$ as given in the question and express $n$ in terms of $x$. We add the latter three integers and equate to $y$as given in the question. We put $n$ in terms of $x$ and simplify to obtain the equation. 

Complete step-by-step solution
We are given in the question that $M$ is a set of six consecutive even integers. We know that even integers are multiples of 2 and two consecutive even integers will have a difference of 2. If we denote an arbitrary even number as $n$, then we can represent the next 5 consecutive even integers in ascending order can be written as $n+2,n+4,n+6,n+6,n+10$. So the set $M$ can be written in listed form as,
$M=\left\{ n,n+2,n+4,n+6,n+8,n+10 \right\}$
We are also given the question that the least three integers of set $M$ are summed, the result is $x$. The least three integers are$n,n+2,n+4$. So we have,
\begin{align} & n+n+2+n+4=x \\ & \Rightarrow 3n+6=x \\ & \Rightarrow n=\dfrac{x-6}{3}.......\left( 1 \right) \\ \end{align}
We are further given the question that when the greatest three integers of set $M$ are summed, the result is $y$.The greatest three integers in the set $M$ are$n+4,n+6,n+10$. So we have,
\begin{align} & n+4+n+6+n+10=y \\ & \Rightarrow 3n+20=y \\ \end{align}
We put $n=\dfrac{x-6}{3}$ obtained from (1) in the above step to have,
\begin{align} & \Rightarrow 3\dfrac{x-6}{3}+24=y \\ & \Rightarrow x-6+24=y \\ & \Rightarrow y=x+18 \\ \end{align}
The above equation is the required equation and hence the correct option is B.

Note: The obtained equation is a linear equation in two variables whose standard form is $ax+by=c$ where $a,b,c$ are real numbers and $a\ne 0,b\ne 0$. There are infinite solutions for one linear equation and we can get integral solutions only when the greatest common divisor of $a,b$ exactly divides $c$. That linear equation is called Diophantine equation.