The sum of two positive numbers is equal to twice the difference between them. If one of the numbers is 6 and the other is [tex]$k$[/tex], where [tex]$k \ \textgreater \ 6$[/tex], then [tex][tex]$k$[/tex][/tex] must satisfy which of the following equations?

A. [tex]6 + k = 2(k - 6)[/tex]

B. [tex]2 + k = 6(k - 2)[/tex]

C. [tex]6 + k = 2(k - 6)[/tex]

D. [tex]6 + k = 6(k - 2)[/tex]



Answer :

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]