Question:
Find the constant of proportionality k. Then write an equation for the relationship between x and y
[tex]\begin{array}{ccccc}x & {2} & {4} & {6} & {8} \ \\ y & {10} & {20} & {30} & {40} \ \ \end{array}[/tex]
Answer:
(a) [tex]k = 5[/tex]
(b) [tex]y = 5x[/tex]
Step-by-step explanation:
Given
[tex]\begin{array}{ccccc}x & {2} & {4} & {6} & {8} \ \\ y & {10} & {20} & {30} & {40} \ \ \end{array}[/tex]
Solving (a): The constant of proportionality:
Pick any two corresponding x and y values
[tex](x_1,y_1) = (2,10)[/tex]
[tex](x_2,y_2) = (6,30)[/tex]
The constant of proportionality k is:
[tex]k = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
[tex]k = \frac{30-10}{6-2}[/tex]
[tex]k = \frac{20}{4}[/tex]
[tex]k = 5[/tex]
Solving (b): The equation
In (a), we have:
[tex](x_1,y_1) = (2,10)[/tex]
k can also be expressed as:
[tex]k = \frac{y- y_1}{x- x_1}[/tex]
Substitute values for x1, y1 and k
[tex]5 = \frac{y- 10}{x- 2}[/tex]
Cross multiply:
[tex]y - 10 = 5(x - 2)[/tex]
Open bracket
[tex]y - 10 = 5x - 10[/tex]
Add 10 to both sides
[tex]y - 10 +10= 5x - 10+10[/tex]
[tex]y = 5x[/tex]
