Question

4) The quantities x and y are in proportion x 4 30 8 __b__ a) b) c) y 6 45 __a__ 15 Find the value of a and b. Find the constant of variation. Is it direct variation or inverse variation?

Answer 1

Question:

The quantities x and y are in proportion

x y

4 6

30 45

8 b

a 15

(a) Find the values of a and b.

(b) Find the constant of variation. Is it direct variation or inverse variation?

Answer:

`a = 10` and
`b = 12`

Constant of variation is 1.5

Direct Variation

Explanation:

Solving (a): The values of a and b

First, we determine the equation that relates x and y

From the table, we have:

`(x_1,y_1) = (4,6)`

`(x_2,y_2) = (30,45)`

The slope (m) is:

`m = (y_2 - y_1)/(x_2 - x_1)`

`m = (45 - 6)/(30 -4)`

`m = (39)/(26)`

`m = 1.5`

The equation is then calculated using:

`y - y_1 = m(x - x_1)`

This gives:

`y - 6 = 1.5(x - 4)`

`y - 6 = 1.5x - 6`

`y = 1.5x`

Solving for the value of b

From the table: x = 8, y = b

Substitute these values in
`y = 1.5x`

`b = 1.5 * 8`

`b = 12`

Solving for the value of a

From the table: x = a, y = 15

Substitute these values in
`y = 1.5x`

`15 = 1.5 * a`

Solve for a

`a = (15)/(1.5)`

`a = 10`

Solving (b): The constant of variation.

This has been solved in (a) above as:

`m = 1.5`

Direct or Inverse?

From the given table, we notice that y increases as x increases and y decreases as x decreases.

This shows direct variation