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.

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

(1 point)
Responses

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

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

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

There is no solution.

Answer 1

There is only one solution.

Answer 2

To determine how many solutions there are to this puzzle, let's solve the system of equations.

Given:
2b + r = 6 -- Equation 1
4b + 2r = 12 -- Equation 2

We can solve this system of equations using the method of substitution or elimination. Let's solve it using the method of substitution.

From Equation 1, we can solve for r in terms of b:
r = 6 - 2b

Substituting this value of r into Equation 2, we get:
4b + 2(6 - 2b) = 12
4b + 12 - 4b = 12
12 = 12

The equation simplifies to 12 = 12, which is always true. This means that the two equations are equivalent and represent the same line on the coordinate plane. As a result, the two equations intersect at every point on this line.

Therefore, there are infinitely many solutions to this puzzle.

Answer 3

To solve this puzzle and determine how many solutions there are, we can use the system of equations provided:

Equation 1: 2b + r = 6 (derived from Clue 1)
Equation 2: 4b + 2r = 12 (derived from Clue 2)

Let's solve this system of equations by using elimination or substitution:

First, let's multiply Equation 1 by 2 to make the coefficients of r the same in both equations:

2(2b + r) = 2(6)
4b + 2r = 12

Now we have:
4b + 2r = 12 (Equation 2)
4b + 2r = 12

Since the equations are the same, there are infinitely many solutions to this puzzle. This means that no matter what values the blue and red numbers are, there will always be a solution that satisfies both clues.

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