Question

Find the constant of proportionality k. Then write an equation for the relationship between x and y

Answer 1

Question:

Find the constant of proportionality k. Then write an equation for the relationship between x and y

`\begin{array}{ccccc}x & {2} & {4} & {6} & {8} \ \\ y & {10} & {20} & {30} & {40} \ \ \end{array}`

Answer:

(a)
`k = 5`

(b)
`y = 5x`

Explanation:

Given

`\begin{array}{ccccc}x & {2} & {4} & {6} & {8} \ \\ y & {10} & {20} & {30} & {40} \ \ \end{array}`

Solving (a): The constant of proportionality:

Pick any two corresponding x and y values

`(x_1,y_1) = (2,10)`

`(x_2,y_2) = (6,30)`

The constant of proportionality k is:

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

`k = (30-10)/(6-2)`

`k = (20)/(4)`

`k = 5`

Solving (b): The equation

In (a), we have:

`(x_1,y_1) = (2,10)`

k can also be expressed as:

`k = (y- y_1)/(x- x_1)`

Substitute values for x1, y1 and k

`5 = (y- 10)/(x- 2)`

Cross multiply:

`y - 10 = 5(x - 2)`

Open bracket

`y - 10 = 5x - 10`

Add 10 to both sides

`y - 10 +10= 5x - 10+10`

`y = 5x`