The perimeter of a rectangle is 74 centimeters. The width is doubled and the length is halved. The new rectangle has a perimeter of 76 centimeters. Find the dimensions.
Okay, so in class we're learning about Systems of Equations. What are the equations for this? How do I define the variables?
2W + 2L = perimeter = 74
2W = 74 -2L
2(2W) + 2(L/2) = 76
Substitute 74 - 2L for 2W in the last equation and solve for L. Put that value in the first equation to find W. Check by putting both values into the last equation.
To solve this problem using systems of equations, let's define two variables to represent the dimensions of the original rectangle:
Let's say:
- x represents the width of the original rectangle, and
- y represents the length of the original rectangle.
We can start by setting up the equations based on the given information:
Equation 1: Perimeter of the original rectangle
The perimeter of a rectangle is given by the formula: P = 2*(length + width)
Since the perimeter of the original rectangle is given as 74 centimeters, the equation becomes:
74 = 2*(x + y)
Equation 2: Perimeter of the new rectangle
According to the problem, the width is doubled and the length is halved for the new rectangle. Hence, the new width is 2x (double the original width), and the new length is y/2 (half of the original length).
Using the same perimeter formula, the perimeter of the new rectangle is given as 76 centimeters:
76 = 2*(2x + y/2)
Now, we have a system of two equations with two variables. We can solve this system to find the values of x and y, which will give us the dimensions of the original rectangle.