Mathematics, 01.07.2019 18:40, Geo777

Prove that b divides a if and only if (-b) divides a

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Because it's if and only if, you need to prove in both directions.

1. prove that if b divides a then -b divides a.
suppose that b divides a. so a = bx (where x∈Z) by definition of "divides".
let x = -1 (-1 is an integer, so we can do this). so a = b(-1), so a = -b.
if a = -b then -b divides a (because any integer divides itself).

2. prove that if -b divides a then b divides a.
suppose that -b divides a. so a = -bx (where x∈Z) by definition of "divides".
a = -x(b) because multiplication is commutative.
let -x = k where k∈Z by closure under multiplication of integers.
a = kb. therefore b divides a by definition of "divides".

thus we have proven that b divides a if and only if (-b) divides a.

i hope this is at least somewhat helpful-- different teachers like different methods/organization.

answered: Guest

a) y = -3x +2

b) y = 1/2x -6

these 2 can be combined into one equation.

-3x +2 = .5x -6

8 = 3.5x

x = 8 / 3.5 = 2.2857142857

y = -3 * (8 / 3.5) +2

y = (-24 / 3.5) + 2

y = -4.8571428571

step-by-step explanation:

answered: Guest

a) 40 more; b) old factory: 10%, new factory: 16%; c) week 4

explanation:

a) at week 0, we see in the graph that the new factory produces 190 specialty items.

using the function, substituting 0 in for w,

$p(w)=230(1.1)^w\\\\p(0)=230(1.1)^0\\\\p(0)=230(1)=230$

this is a difference of 230-190 = 40.

b) the function for the old factory is written in the form f(x) = a(1+r)ˣ, where a is the original amount, x is the amount of time and r is the growth rate. the equation for the old factory is $p(w)=230(1.1)^w$ ; 230 is in place of a and is the original amount; w is the amount of time; which means 1+r = 1.1. this means that r must be 0.1 or 10%.

for the new factory, we find the percent of increase to find the growth rate; the formula for this is

$\text{percent of change}=\frac{\text{amount of change}}{\text{original amount}}$

between the first two points, the amount of change is 220-190 = 30; this gives us

30/190 = 0.158 ≈ 0.16 = 16%

c) at week 0, the old factory has 230 while the new factory has 190.

at week 1, the old factory has 230(1.1) = 253 while the new factory has 220.

at week 2, the old factory has 230(1.1²) = 278.3 while the new factory has 252.

at week 3, the old factory has 230(1.1³) = 306.13 while the new factory has 290.

at week 4, the old factory has 230(1.1⁴) = 336.743 while the new factory has 337. this is the first week that the new factory surpasses the old.

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