You are trying to guess the code to a puzzle while participating in an escape room. There are two buttons, blue and red, and you have to put the correct number in each one to unlock the door to the next room. There are only two clues to help you:

Question

You are trying to guess the code to a puzzle while participating in an escape room. There are two buttons, blue and red, and you have to put the correct number in each one to unlock the door to the next room. There are only two clues to help you:

• Clue 1: If you double the blue number and add it to the red number, you get 6.
• Clue 2: If you quadruple the blue number and add it to double the red number, you get 12.
You only get two attempts at the puzzle, and if you get it wrong, the game will be over. Use the system of equations shown to determine how many solutions there are to this puzzle.
{26 + = 6
46+ 2r =12point)
• There are only two solutions.
• There is only one solution.
• There is no solution.
• There are infinitely many solutions.

Answer 1

Bot GPT 4-o mini answered
2 hours ago

To solve the puzzle, let's define our variables based on the clues given:

Let:

\( b \) be the number associated with the blue button.
\( r \) be the number associated with the red button.

From the clues provided, we can establish the following equations:

From Clue 1: \( 2b + r = 6 \)
From Clue 2: \( 4b + 2r = 12 \)

Now, we can simplify the second equation by dividing everything by 2:

\[ 2b + r = 6 \]

Now we have two identical equations:

\( 2b + r = 6 \)
\( 2b + r = 6 \)

Since both equations are the same, they represent the same line in a graph. This means they are dependent equations, leading to infinitely many solutions depending on the values of \( b \) and \( r \) that satisfy this equation.

In conclusion, the answer is:

There are infinitely many solutions.