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 no solution.
There is no solution.

There are only two solutions.
There are only two solutions.

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

There is only one solution.

Answer 1

To solve the system of equations provided by the clues, we will set up the equations based on the clues given:

From Clue 1: \[ 2b + r = 6 \] (1)

From Clue 2: \[ 4b + 2r = 12 \] (2)

Now let's simplify Equation (2). We can divide the entire equation by 2: \[ 2b + r = 6 \] (3)

Notice that Equation (3) is actually the same as Equation (1).

Since both equations represent the same line, they have infinitely many solutions. Any point on this line will satisfy both equations.

Thus, the answer is:

There are infinitely many solutions.