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)ResponsesThere 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. step by step bot
To solve this puzzle, we can set up a system of equations based on the given clues.
Let's define the variables:
b = blue number
r = red number
From clue 1, we have the equation:
2b + r = 6
From clue 2, we have the equation:
4b + 2r = 12
Now, we can solve this system of equations.
Step 1: Multiply the first equation by 2 to eliminate the variable r:
2(2b + r) = 2(6)
4b + 2r = 12
Step 2: Compare the result with the second equation:
4b + 2r = 12
Both equations are the same, which means they represent the same line. Therefore, there are infinitely many solutions to this puzzle.
Answer: There are infinitely many solutions.
To solve this puzzle, we can set up a system of equations based on the given clues.
Let's assume the blue number is represented by "b" and the red number is represented by "r".
Clue 1: If you double the blue number and add it to the red number, you get 6.
This can be written as the equation: 2b + r = 6.
Clue 2: If you quadruple the blue number and add it to double the red number, you get 12.
This can be written as the equation: 4b + 2r = 12.
Now, we have a system of equations:
2b + r = 6 ----- (Equation 1)
4b + 2r = 12 ----- (Equation 2)
To solve the system, we can use the method of substitution or elimination. Let's use the method of elimination.
Multiplying Equation 1 by 2, we get:
4b + 2r = 12 ----- (Equation 3)
Now, we can subtract Equation 2 from Equation 3:
(4b + 2r) - (4b + 2r) = 12 - 12
0 = 0
Since 0 = 0, this implies that the two equations are actually equivalent.
Therefore, we can conclude that there are infinitely many solutions to this puzzle.
To solve the puzzle and determine how many solutions there are, we need to solve the system of equations provided.
The system of equations is:
2b + r = 6 (Equation 1)
4b + 2r = 12 (Equation 2)
To solve the system, we can use the method of elimination or substitution. I will show you the elimination method.
Step 1: Multiply Equation 1 by 2 to make the coefficient of b equal in both equations:
2(2b + r) = 2(6)
This simplifies to:
4b + 2r = 12 (Equation 3)
Step 2: Subtract Equation 2 from Equation 3:
(4b + 2r) - (4b + 2r) = (12) - (12)
This simplifies to:
0 = 0
Step 3: Analyzing the result:
We obtained a true statement, 0 = 0. This means that the two equations are actually the same equation. In other words, Equation 1 is a multiple of Equation 2. Therefore, the two equations describe the same line in the coordinate plane, and there are infinitely many solutions to this system.
In the context of the puzzle, it means that there are infinitely many combinations of blue and red numbers that satisfy both clues and unlock the door to the next room. You can choose any valid combination, and it will be a correct solution.
So, the answer is: There are infinitely many solutions to this puzzle.