Let's solve the given problem step-by-step:
We are given several equations and clues related to Δ:
1. [tex]\( \Delta + \Delta + \dot{\Delta} = 22 \)[/tex]
2. [tex]\( \Delta \div 2 = 4 \)[/tex]
3. [tex]\( 2 + \Delta + \Delta = \Delta \)[/tex]
First, let’s start with the second equation to find Δ:
[tex]\[ \Delta \div 2 = 4 \][/tex]
To solve for Δ, we multiply both sides of the equation by 2:
[tex]\[ \Delta = 4 \times 2 \][/tex]
[tex]\[ \Delta = 8 \][/tex]
Now that we know [tex]\( \Delta = 8 \)[/tex], let’s verify that it satisfies the given clues.
First clue: [tex]\( 2 + \Delta + \Delta = \Delta \)[/tex]
Let’s substitute [tex]\( \Delta = 8 \)[/tex] into this clue:
[tex]\[ 2 + 8 + 8 = \Delta \][/tex]
[tex]\[ 18 = 8 \][/tex]
This equation does not hold true, so the first clue is false.
Now let’s check the original equation:
[tex]\[ \Delta + \Delta + \dot{\Delta} = 22 \][/tex]
[tex]\[ 8 + 8 + \dot{\Delta} = 22 \][/tex]
[tex]\[ 16 + \dot{\Delta} = 22 \][/tex]
Solving for [tex]\( \dot{\Delta} \)[/tex]:
[tex]\[ \dot{\Delta} = 22 - 16 \][/tex]
[tex]\[ \dot{\Delta} = 6 \][/tex]
Finally, let's verify the second clue:
[tex]\[ \Delta \div 2 = 4 \][/tex]
[tex]\[ 8 \div 2 = 4 \][/tex]
This equation holds true, so the second clue is true.
To summarize:
- [tex]\( \Delta = 8 \)[/tex]
- [tex]\( \dot{\Delta} = 6 \)[/tex]
- First clue is false: [tex]\( 2 + \Delta + \Delta = \Delta \)[/tex] does not hold as [tex]\( 18 \neq 8 \)[/tex]
- Second clue is true: [tex]\( \Delta \div 2 = 4 \)[/tex] holds as [tex]\( 8 \div 2 = 4 \)[/tex]
Thus, the final solutions are:
- [tex]\( \Delta = 8 \)[/tex]
- [tex]\( \dot{\Delta} = 6 \)[/tex]
