Let the two consecutive even integers be represented as \( x \) (the lesser integer) and \( x + 2 \) (the greater integer).
According to the problem, we can set up the following equation based on the condition given:
\[ 6x + (x + 2) = 86 \]
Now, we simplify the equation:
\[ 6x + x + 2 = 86 \]
Combine like terms:
\[ 7x + 2 = 86 \]
Next, subtract 2 from both sides to isolate the term with \( x \):
\[ 7x = 86 - 2 \]
\[ 7x = 84 \]
Now, divide both sides by 7 to solve for \( x \):
\[ x = \frac{84}{7} = 12 \]
Now that we have the value of \( x \), we can find the two consecutive even integers:
The lesser integer is \( x = 12 \), and the greater integer is:
\[ x + 2 = 12 + 2 = 14 \]
Thus, the two consecutive even integers are \( 12 \) and \( 14 \).
To verify, we can check the original condition:
Calculate \( 6 \times 12 + 14 \):
\[ 6 \times 12 = 72 \] \[ 72 + 14 = 86 \]
Since our calculation holds true, the solution is confirmed. Therefore, the two consecutive even integers are
\[ \boxed{12} \text{ and } \boxed{14}. \]