Question

Bruce retiled his kitchen. Originally, he bought three boxes of tile with the same number of tiles in each. He ran out of tile and had to go back to get three extra boxes of tile. However, he only used two tiles from the last box to finish the job. Felicia also retiled her kitchen. She bought five boxes of tile, and each box contained the same number of tiles as each of Bruce’s boxes. She also ran out of tile. She had to go back to the store and get five more tiles to finish the job. Write an expression for the number of tiles Bruce used and an expression for the number of tiles Felicia used. Use x to represent the number of tiles in a box. To determine whether they used the same number of tiles, set the two expressions equal to each other and solve for x. Is there a solution for x? Why or why not? If there is a solution, what is it? What conclusion can you draw from this? How could you change the equation from Part B so it has infinitely many solutions? What would infinitely many solutions mean in terms of the situation? If the equation were 6x + 2 = 5x + 17, would there be one unique solution? What is it, and what would it mean in terms of the situation?

Answer 1

Bruce's expression for tiles used is 3x + 2, while Felicia's is 5x + 5. Equating and solving shows no solution exists for x, which means they didn't use the same number of tiles. The equation 6x + 2 = 5x + 17 has a unique solution of x = 15, representing the number of tiles in a box.

Step-by-step explanation:

The student is asking to create mathematical expressions for the number of tiles Bruce and Felicia used and then to compare them to determine if they used the same number of tiles. Let's denote the number of tiles in a box as x.

Bruce bought 3 boxes and used 2 extra tiles, so the expression for the number of tiles he used is: 3x + 2.

Felicia bought 5 boxes and needed 5 extra tiles, so her expression is: 5x + 5.

Setting the expressions equal to find out if they used the same amount:

3x + 2 = 5x + 5.

Solving for x gives us no solution because when we try to isolate x, we end up with an equation that is not possible, such as 0 = 3, which indicates that they cannot have used the same number of tiles with this equation.

To make the equation have infinitely many solutions, we would need both sides to be identical, which would represent them using the same amount of tile regardless of the box count. An example would be 3x + 2 = 3x + 2.

If the equation were 6x + 2 = 5x + 17, there would indeed be one unique solution. By solving we get x = 15, which suggests that a box contains 15 tiles, and the equation would represent a situation where Bruce and Felicia used different numbers of tiles.