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

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.

{2b+r=64b+2r=12

(1 point)
Responses

There is only one solution.
There is only one solution.

There are infinitely many solutions.
There are infinitely many solutions.

There is no solution.
There is no solution.

There are only two solutions.

Answer 1

To solve the puzzle, let’s define the blue number as \( b \) and the red number as \( r \). We have the following system of equations based on the clues provided:

1. \( 2b + r = 6 \) (from Clue 1)
2. \( 4b + 2r = 12 \) (from Clue 2)

We can simplify the second equation by dividing everything by 2:

\[ 2b + r = 6 \]

Now, both equations are actually the same:

1. \( 2b + r = 6 \)
2. \( 2b + r = 6 \)

Since both equations represent the same line, they do not provide independent information. This means there are infinitely many solutions that satisfy this system because any value of \( b \) will produce a corresponding \( r \) that satisfies the equation.

Conclusion

The answer is: There are infinitely many solutions.