How do you find three consecutive even integers such that three times the largest is 34 more than the sum of the two smaller integers?

Hint: To find three consecutive even integers such that three times the largest is 34 more than the sum of the two smaller integers. Let us consider the smallest of the three consecutive even integers to be x. Then, we can write the next two consecutive even integers as $x+2,x+4$ . Now, let us denote the three times the largest integer as $3\left( x+4 \right)$ . We are given that three times the largest is 34 more than the sum of the two smaller integers. This can be written as $3\left( x+4 \right)=34+x+\left( x+2 \right)$ . By solving for x, we can get all the three consecutive even integers.

Complete step-by-step solution:
We need to find three consecutive even integers such that three times the largest is 34 more than the sum of the two smaller integers. Let us consider the smallest of the three consecutive even integers to be x.
Then, we can write the next two consecutive even integers as $x+2,x+4$ .
Now, let us write the three largest integers.
$\Rightarrow 3\left( x+4 \right)$
We are given that three times the largest is 34 more than the sum of the two smaller integers. This can be written as
$3\left( x+4 \right)=34+x+\left( x+2 \right)$
Let us apply distributive property in the LHS.
$\Rightarrow 3x+12=34+x+x+2$
Let us now solve the RHS. We will get
$\Rightarrow 3x+12=36+2x$
We have to collect the x terms on one side and constants on the other side.
$\Rightarrow 3x-2x=36-12$
When we solve the above expression, we get
$\Rightarrow x=24$
Thus the smallest even integer is 24.
Let us find the next consecutive even integer, that is, $x+2$ .
$x+2=24+2=26$
Now, we can find the largest consecutive even integer.
$x+4=24+4=28$
Hence, the three consecutive even integers are 24,26 and 28.

Note: Students must be aware that when writing consecutive even integers, they must add 2,4,6,..etc. They may also commit mistakes by ending the solution after finding x and thus skips to find the consecutive numbers.