A golden ratio problem

Question

How do I solve these two equations for the golden ratio problem?

So the problem is about a straw. You're supposed to cut it approximately in half then write a system of equations.

For the system of equations I have x+y=7 and the other is 7/y=y/x

We are supposed to solve the system and then use it to find the golden mean. Any help is appreciated even for just the first part!

Answer 1

As an addendum to Andrew's solution, I should say that the golden ratio/mean is equal to the length of the longer piece to the length of the whole straw.

 

The longer piece is, as Andrew gives, [-7 + √185 ] / 2 , which we can simplify to [ -7 + 7 * √5 ] / 2 .

 

So when we divide by 7 (the length of the whole straw), we should get [ -1 + √5 ] / 2, which is the golden ratio, commonly denoted with the Greek letter phi, φ. It's approximately equal to 1.618, and it has some pretty interesting properties. For example, φ-1 = 1 / φ. 

Answer 2

Apparently we have a 7 inch long straw that is cut into two pieces.

 

1)  x+y = 7

2)  7/y = y/x

 

From equation 1 we can solve for x

x = 7-y

 

cross multiply on equation 2

y² = 7x

substitute 7-y in place of x

y² = 7(7-y)

y² = 49-7y

y² + 7y - 49 = 0

This is a quadratic equation and can be solved

using the quadratic equation.

y = [-b±√(b²-4ac)]/2a    a=1, b=7, c=-49

 

y = [-7 ±√(7²-4(-49)]/2

y = [-7 ±√185]/2

y = (-7 ± 13.6/2)

y = -20.6/2 or 6.6/2

y = -10.3 or 3.3

 

Since the length of the piece cannot be negative

discard the -10.3

 

y = 3.3

x = 7-3.3 = 3.7

 

Your two pieces are length 3.3 and 3.7