Question
Find two consecutive even integers such that six times the lesser added to the greater gives a sum of 86
Answers
GPT-4o mini
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
\[
\boxed{12} \text{ and } \boxed{14}.
\]
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
\[
\boxed{12} \text{ and } \boxed{14}.
\]
Laniilsia
RU sure?
GPT-4o mini
Yes, let's go through the solution again step by step to ensure accuracy.
We want to find two consecutive even integers such that six times the lesser added to the greater gives a sum of 86.
**Step 1: Define the integers.**
Let the lesser even integer be \( x \). Therefore, the next consecutive even integer would be \( x + 2 \).
**Step 2: Set up the equation.**
According to the problem, we need to set up the equation based on the condition provided:
\[
6x + (x + 2) = 86
\]
**Step 3: Simplify the equation.**
Combine the terms:
\[
6x + x + 2 = 86
\]
This simplifies to:
\[
7x + 2 = 86
\]
**Step 4: Solve for \( x \).**
Subtract 2 from both sides:
\[
7x = 86 - 2
\]
\[
7x = 84
\]
Now, divide both sides by 7:
\[
x = \frac{84}{7} = 12
\]
**Step 5: Find the two integers.**
The lesser integer is \( x = 12 \), and the greater integer is:
\[
x + 2 = 12 + 2 = 14
\]
**Step 6: Verify the solution.**
Now let's check whether the condition is satisfied:
Calculate \( 6 \times 12 + 14 \):
\[
6 \times 12 = 72
\]
\[
72 + 14 = 86
\]
Since the left-hand side equals the right-hand side, our solution is indeed correct.
Therefore, the two consecutive even integers are
\[
\boxed{12} \text{ and } \boxed{14}.
\]
We want to find two consecutive even integers such that six times the lesser added to the greater gives a sum of 86.
**Step 1: Define the integers.**
Let the lesser even integer be \( x \). Therefore, the next consecutive even integer would be \( x + 2 \).
**Step 2: Set up the equation.**
According to the problem, we need to set up the equation based on the condition provided:
\[
6x + (x + 2) = 86
\]
**Step 3: Simplify the equation.**
Combine the terms:
\[
6x + x + 2 = 86
\]
This simplifies to:
\[
7x + 2 = 86
\]
**Step 4: Solve for \( x \).**
Subtract 2 from both sides:
\[
7x = 86 - 2
\]
\[
7x = 84
\]
Now, divide both sides by 7:
\[
x = \frac{84}{7} = 12
\]
**Step 5: Find the two integers.**
The lesser integer is \( x = 12 \), and the greater integer is:
\[
x + 2 = 12 + 2 = 14
\]
**Step 6: Verify the solution.**
Now let's check whether the condition is satisfied:
Calculate \( 6 \times 12 + 14 \):
\[
6 \times 12 = 72
\]
\[
72 + 14 = 86
\]
Since the left-hand side equals the right-hand side, our solution is indeed correct.
Therefore, the two consecutive even integers are
\[
\boxed{12} \text{ and } \boxed{14}.
\]