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.

Answer 1

To solve this puzzle, we need to translate the clues into a system of equations. Let's denote the blue number as \( b \) and the red number as \( r \).

From Clue 1, we have:

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

From Clue 2, we have:

2. \( 4b + 2r = 12 \)

Let's simplify the second equation. We can divide the entire equation by 2:

\[ 2b + r = 6 \]

Now we have:

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

Notice that both equations are identical. This means that they represent the same line when graphed, and thus they have an infinite number of solutions along that line.

To express \( r \) in terms of \( b \) from either equation, we can rearrange Equation 1:

\[ r = 6 - 2b \]

We have expressed \( r \) in terms of \( b \), which indicates that for any value of \( b \), we can find a corresponding \( r \). For example, if \( b = 0 \), then \( r = 6 - 2 \cdot 0 = 6 \). If \( b = 1 \), then \( r = 6 - 2 \cdot 1 = 4\), and so on.

Thus, there are infinitely many pairs of \( (b, r) \) that satisfy the equations.

In conclusion: There are an infinite number of solutions to this puzzle based on the given clues.