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.

Question

{2b+r=6
{4b+2r=12 (1 point)

There are only two solutions. There are only two solutions. There is only one solution. There is only one solution. There is no solution. There is no solution. There are infinitely many solutions

Answer 1

To solve the system of equations:

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

Let's simplify Equation 2. Notice that we can divide every term in Equation 2 by 2:

\[ \frac{4b + 2r}{2} = \frac{12}{2} \]

This simplifies to:

\[ 2b + r = 6 \]

As you can see, Equation 2 is actually the same as Equation 1. This means that both equations represent the same line or are essentially the same equation in two variables.

Since both equations are identical, there are infinitely many solutions along the line defined by the equation \( 2b + r = 6 \).

To summarize the results:

• There are infinitely many solutions because the system of equations is dependent (the second equation is a multiple of the first).

Thus, the answer is: There are infinitely many solutions.