There are 3 consecutive even integers. if twice the first integer added to the third is 268,216,find all three integers

We can use a system of equations to solve this.

Notice that we are required to find 3 consecutive even integers.

This can be represented as x, y, and z, where x is the smallest integer, y is the next largest, and z is the largest of the three.
x = x
y = x + 2
z = y + 2

Take a moment to make sense of the above system of equations.
I can simplify the system a bit more by plugging in the second equation into the third one.

z = (x + 2) + 2
z = x + 4

Now the new system of equations is 
x = x
y = x + 2
z = x + 2

Now for the second part of this problem. We are given this equation
2x + z = 268216

You may notice that we no longer need the second equation, y = x + 2. Also, the first equation is redundant, so we can ignore it.

That leaves us with this system:
z = x + 4
2x + z = 268216

What do you think we should do? We can plug in z to solve for x.

2x + (x + 4) = 268216
3x + 4 = 268216
3x = 268212
x = 89404

Now we know the first number in the sequence! Then we can find the other two numbers, because we know that the next numbers must be the next largest even numbers in sequence!
89404, 89406, and 89408 are your answers.