Answer
1)
a) b = 9
b) y = 15
c) w = 5
2)
a) p = 9
b) y = -14
c) Z =
Step-by-step explanation
1)
a) We are told that the variable pairs are proportional to each other, that is, b varies directly as a, which can be written as
b ∝ a
Introducing the constant of variation, k, we have
b ∝ a
b = ka
We can then solve for k knowing that
b = 8 when a = 16
8 = 16k
16k = 8
(16k/16) = (8/16)
k = 0.5
when a = 18,
b = ka
b = 0.5a
b = 0.5 (18) = 9
b) y = 10 when x = 14, what is y when x = 21?
y = kx
10 = 14k
14k = 10
(14k/14) = (10/14)
k = (5/7)
y = (5x/7)
y = (5 × 21)/7 = 15
c) w = -2 when u = 6, w = ? when u = -15
w = ku
-2 = 6k
6k = -2
(6k/6) = (-2/6)
k = (-1/3)
w = (-u/3)
w = (-(-15)/3)
w = (15/3) = 5
2) For these ones, the variables vary directly as each other.
p ∝ q
Introducing the constant of variation, k, we have
p ∝ q
p = kq
We can then solve for k knowing that
p = 12 when q = 8
p = kq
12 = 8k
8k = 12
(8k/8) = (12/8)
k = (3/2)
p = (3q/2)
p = (3 × 6)/2 = 9
b) y = 21 when x = 9, what is y when x = -6?
y = kx
21 = 9k
9k = 21
(9k/9) = (21/9)
k = (7/3)
y = (7x/3)
y = (7 × -6)/3 = -14
c) Z = -5 when w = 2, what is Z when w = 8?
Z = kw
-5 = 14k
14k = 10
(14k/14) = (10/14)
k = (5/7)
y = (5x/7)
y = (5 × 21)/7 = 15