If one side of a square is doubled in length and the adjacent side is decreased by two centimeters, the area of the resulting rectangle is 96 square centimeters larger than that of the original square. Find the dimensions of the rectangle.
I'm starting from a square with sides of some unknown length. The sides of the rectangle are defined in terms of that unknown length. So I'll pick a variable for the unknown side-length, create expressions for the rectangle's sides, and then work from there. square's side length: x one side is doubled: 2x next side is decreased by two: x – 2 square's area: x2 rectangle's area: (2x)(x – 2) = 2x2 – 4x new area is 96 more than old area: 2x2 – 4x = x2 + 96 2x2 – 4x = x2 + 96 x2 – 4x – 96 = 0 (x – 12)(x + 8) = 0 x = 12 or x = –8 I'm supposed to find the dimensions of a rectangle, so can I just erase that one "minus" sign and say that the rectangle is 12 by 8? No! I defined "x" as standing for the side length of the square, not as one of the sides of the rectangle. Looking back at my definitions, I see what "x = 12" (the only reasonable solution for the square) means that the sides of the rectangle have lengths 2(12) and (12) – 2: The rectangle measures 24 cm by 10 cm.
If you wanted to find the empty cells within a range, what formula would work?
For what value of the constant K does the system of equations 2x - y = 4 and 6x - 3y = 3K have an infinite number of solutions?
For the system of equations 2x - y = 4 and 6x - 3y = 3K to have an infinite number of solutions, the two equations must be equivalent. If we multiply all terms of the first equation by 3, we obtain 6x - 3y = 12 For the two equations to be equivalent we must have 12 = 3K which when solved gives K = 4.