Alvita is planning a garden patio to be made from identical square paving stones laid out in a rectangle measuring 2 stones by y stones.
She finds that when she adds a border of width one stone around the patio, the area of the border is equal
to the original area of the patio.
How many possible values for x are there?

Step-by-step explanation:

what means "x" ?

anyway, we have an original rectangle

2 stones × y stones with the area of 2y.

adding a border of one line of stones around that requires

3 stones on both sides (2 stones next to the existing ones, and 1 stone for the extra row above the existing 2 rows) and another row of y stones.

that are 2×3 + y = 6 + y stones.

the area of a 1 stone thick border is exactly also the number of used stones = 6 + y.

and now we know that the original area (2y) is the same as the area of the new border.

so,

2y = 6 + y

y = 6

there is only one possible solution, as this is a linear equation (polynomial of first or 1 degree).

he used the algebra tiles correctly.

step-by-step explanation:

given : 3x(2x-1)

solution :

since he used algebra tiles .

armando's solution is the same in the solution shown in figure

as he distributed 2x into x and x and wrote them vertically

and also write   -1 vertically

and 3x into x and x and x and wrote it horizontally

performing the multiplication in the algebra tiles we get

and solving the given expression :

3x(2x-1)

since solving the expression using tiles   and solving the expression algebraically we get the same answer

thus   we can choose the first statement as the answer to this problem which is "he used the algebra tiles correctly".