Let's solve the problem step-by-step.
Given:
- One of the numbers is [tex]\( 6 \)[/tex]
- The other number is [tex]\( k \)[/tex]
- [tex]\( k > 6 \)[/tex]
- The sum of the two numbers is equal to twice the difference between them
First, we express the given conditions as equations:
1. The sum of the numbers is [tex]\( 6 + k \)[/tex]
2. The difference between the numbers is [tex]\( k - 6 \)[/tex] (since [tex]\( k > 6 \)[/tex])
3. The sum equals twice the difference:
Therefore,
[tex]\[ 6 + k = 2 \times (k - 6) \][/tex]
Now, let's solve this equation:
[tex]\[ 6 + k = 2(k - 6) \][/tex]
Start by expanding the right-hand side:
[tex]\[ 6 + k = 2k - 12 \][/tex]
Next, let's isolate [tex]\( k \)[/tex] on one side. Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[ 6 = k - 12 \][/tex]
Now, add 12 to both sides to solve for [tex]\( k \)[/tex]:
[tex]\[ 6 + 12 = k \][/tex]
[tex]\[ k = 18 \][/tex]
So, the correct equation that [tex]\( k \)[/tex] must satisfy is:
[tex]\[ 6 + k = 2(k - 6) \][/tex]
Therefore, the correct answer is:
[tex]\[ 6 + k = 2(k - 6) \][/tex]
Which corresponds to the option:
[tex]\[ \boxed{6+k=2(k-6)} \][/tex]
