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:
[tex]a = 10[/tex] and [tex]b = 12[/tex]
Constant of variation is 1.5
Direct Variation
Step-by-step explanation:
Solving (a): The values of a and b
First, we determine the equation that relates x and y
From the table, we have:
[tex](x_1,y_1) = (4,6)[/tex]
[tex](x_2,y_2) = (30,45)[/tex]
The slope (m) is:
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
[tex]m = \frac{45 - 6}{30 -4}[/tex]
[tex]m = \frac{39}{26}[/tex]
[tex]m = 1.5[/tex]
The equation is then calculated using:
[tex]y - y_1 = m(x - x_1)[/tex]
This gives:
[tex]y - 6 = 1.5(x - 4)[/tex]
[tex]y - 6 = 1.5x - 6[/tex]
[tex]y = 1.5x[/tex]
Solving for the value of b
From the table: x = 8, y = b
Substitute these values in [tex]y = 1.5x[/tex]
[tex]b = 1.5 * 8[/tex]
[tex]b = 12[/tex]
Solving for the value of a
From the table: x = a, y = 15
Substitute these values in [tex]y = 1.5x[/tex]
[tex]15 = 1.5 * a[/tex]
Solve for a
[tex]a = \frac{15}{1.5}[/tex]
[tex]a = 10[/tex]
Solving (b): The constant of variation.
This has been solved in (a) above as:
[tex]m = 1.5[/tex]
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
