Question

Two variables, y and x, are inversely proportional.

When x = 12, y = 10.

(a) Find an equation that relates y and x.

(b) Given that x and y are positive integers, find the values of x and y when y=x+7.

I don’t understand this question at all!

Answer 1

x = 8, y = 15

Explanation:

Part (a)

If x and y are inversely proportional, the relationship can be expressed as

`y \propto (1)/(x)\\\\\rm{(or \;alternatively\;x \propto (1)/(y))\\\\}\\\\`

The above relationship
`y \propto (1)/(x)` can be written as an equation:

`y = (k)/(x) \codts\cdots(1)`

where the constant k is known as the constant of proportionality

Part (b)

We know that when x = 12, y = 10

Plugging this into equation (1):

10 = k/12

k/12 = 10

k = 12 x 10

k = 120

So the proportional equation (1) becomes
y = 120/x

We are now given the equation
y = x + 7

Substitute y = 120/x

120/x = x + 7

Subtract 120/x from both sides
0 = x + 7 - 120/x

Multiply by x both sides;
0 = x² + 7x - 120

Or,
x² + 7x - 120 = 0

This is a quadratic equation which we can solve by factoring. There are various techniques. One of them is to find two factors of 120 and see if their sum or difference can be made -7

Factors of 120 are:

`1,\:2,\:3,\:4,\:5,\:6,\:8,\:10,\:12,\:15,\:20,\:24,\:30,\:40,\:60,\:120`

Take the two factors, a and b that will add or subtract to -7 and multiply to -120

We see that if we choose a = -15 and b = 8

a + b = -7

and

a x b = -120

The solution to the equation is x = 8 or x = -15

Since we are told that both x and y are positive, we can ignore x = -15 and just state that x = 8 is a solution

If x = 8, substitute this in y = x + 7 to get

y = 8 + 7 = 15

This checks with our original proportionality equation:
y = 120/x = 120/8 = 15

Answer
x = 8, y = 15