Billy is framing a picture with a rectangular frame. The frame has a perimeter of 62 inches. The height of the frame is 2⁄3 times the length. Find the dimensions of the rectangle.

a. Define the variable(s)
b. Write an equation to represent the situation.
c. Solve.

im stuck on how to start this

2 h + 2 L = 62

so
L = 31 - h

h = 2 L/3
so
L = 1.5 h

1.5 h = 31 - h
2.5 h = 31
h = 12.4
L = 18.6

i understand most of this, but I'm a bit confused on this part:

h = 2L/3
so
L = 31 - h

how did you get this?

if 2 L + 2 h = 62 given

then L + h = 31
and L = 31 - h

if h = 2L/3 given
then L = 3 h/2 = 1.5 h

To start solving the problem, you can follow these steps:

a. Define the variable(s):
Let's define the variable "L" to represent the length of the rectangle.

b. Write an equation to represent the situation:
Given that the frame has a perimeter of 62 inches, we can express this mathematically using the formula for the perimeter of a rectangle:
Perimeter = 2 * (Length + Width)

Since the height (H) of the frame is 2/3 times the length (L), we can write:
Height (H) = (2/3) * Length (L)

Now, we can substitute the values into the perimeter equation to get the equation that represents the situation:
62 = 2 * (L + (2/3)L)

c. Solve:
To solve the equation, we can simplify it step by step:

62 = 2 * (L + (2/3)L)
62 = 2 * (L + 2L/3) [Distributed 2 to both terms inside the brackets]
62 = 2 * (3L/3 + 2L/3) [Combined L and 2L/3 by finding a common denominator]
62 = 2 * (5L/3) [Combined the terms inside the brackets]
62 = (10/3)L [Multiplied 2 by 5L/3]
(62 * 3) / 10 = L [Divided both sides by 10/3 to isolate L]
186/10 = L [Simplified the multiplication]
18.6 = L [Divided 186 by 10]

Therefore, the length of the rectangle is 18.6 inches.

To find the height, we can substitute the value of L into the equation for H:

Height (H) = (2/3) * Length (L)
Height (H) = (2/3) * 18.6
Height (H) = 12.4 inches

So, the dimensions of the rectangle are 18.6 inches by 12.4 inches.