To determine which equation is true based on the given facts, let's analyze the problem step-by-step.
### Given Facts:
1. Mark (m) weighs twice as much as Kelly (k).
2. Mark (m) weighs 50 pounds more than Vincent (v).
### Translations into Equations:
1. Mark weighs twice as much as Kelly: [tex]\( m = 2k \)[/tex]
2. Mark weighs 50 pounds more than Vincent: [tex]\( m = v + 50 \)[/tex]
### Step-by-Step Solution:
1. From the first fact, we have the equation:
[tex]\[
m = 2k
\][/tex]
2. From the second fact, we have another equation:
[tex]\[
m = v + 50
\][/tex]
3. To find a relation between [tex]\( k \)[/tex] and [tex]\( v \)[/tex], we can substitute the expression for [tex]\( m \)[/tex] from the first equation into the second equation. So, we substitute [tex]\( m = 2k \)[/tex] into [tex]\( m = v + 50 \)[/tex]:
[tex]\[
2k = v + 50
\][/tex]
### Comparisons with Given Options:
- F. [tex]\( 2m + v = 50 \)[/tex]
Let's check this with our derived equations:
[tex]\[
2(2k) + v = 50 \Rightarrow 4k + v = 50
\][/tex]
This equation does not align with [tex]\( 2k = v + 50 \)[/tex].
- G. [tex]\( m + v = 2k \)[/tex]
We check with [tex]\( m = 2k \)[/tex]:
[tex]\[
2k + v = 2k
\][/tex]
Subtracting [tex]\( 2k \)[/tex] from both sides:
[tex]\[
v = 0
\][/tex]
This is not consistent with the given facts.
- H. [tex]\( 2v + k = 50 \)[/tex]
Substitute [tex]\( v = 2k - 50 \)[/tex]:
[tex]\[
2(2k - 50) + k = 50 \Rightarrow 4k - 100 + k = 50 \Rightarrow 5k - 100 = 50 \Rightarrow 5k = 150 \Rightarrow k = 30
\][/tex]
Assuming [tex]\( k = 30 \)[/tex]:
[tex]\[
v = 2(30) - 50 = 60 - 50 = 10
\][/tex]
Valid but needs to hold for all values.
- J. [tex]\( 2k + 50 = v \)[/tex]
Substitute [tex]\( m = 2k \)[/tex]:
This does not match our derived equation [tex]\( 2k = v + 50 \)[/tex].
- K. [tex]\( 2k = v + 50 \)[/tex]
This directly matches our derived equation:
[tex]\[
2k = v + 50
\][/tex]
### Conclusion:
The correct equation that aligns with both given facts is:
[tex]\[
2k = v + 50
\][/tex]
Therefore, the correct answer is K.
