From 6,1 to 7,4
We can determine the slope of the line
rise/run
3/1=3

Now we know the slope of the line, a line parallel to it must obtain the same slope, therefore, I tried to see if the increase in y is three times the increase in x
I found the true pair to be A

(-4,1) and (-2,7)

6 0

3 years ago

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At a train station, freight trains depart at a rate of 2 trains every 30 minutes and passenger trains depart at a rate of 1 trai

never [62]

In my work I used t=trains and m=minutes:
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
30+30= 60m/2t
Total= 720m/24t

1:30+1:30+1:30+1:30+1:30+1:30+1:30+1:30+1:30+1:30+1:30+1:30= 4 o' clock
Total= 372m/12t

24t+12t= 36t

The answer is:
36 trains in total

6 0

3 years ago

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false:

kondaur [170]

$\text{Proof by induction:}$
$\text{Test that the statement holds or n = 1}$

$LHS = (3 - 2)^{2} = 1$
$RHS = \frac{6 - 4}{2} = \frac{2}{2} = 1 = LHS$
$\text{Thus, the statement holds for the base case.}$

$\text{Assume the statement holds for some arbitrary term, n= k}$
$1^{2} + 4^{2} + 7^{2} + ... + (3k - 2)^{2} = \frac{k(6k^{2} - 3k - 1)}{2}$

$\text{Prove it is true for n = k + 1}$
$RTP: 1^{2} + 4^{2} + 7^{2} + ... + [3(k + 1) - 2]^{2} = \frac{(k + 1)[6(k + 1)^{2} - 3(k + 1) - 1]}{2} = \frac{(k + 1)[6k^{2} + 9k + 2]}{2}$

$LHS = \underbrace{1^{2} + 4^{2} + 7^{2} + ... + (3k - 2)^{2}}_{\frac{k(6k^{2} - 3k - 1)}{2}} + [3(k + 1) - 2]^{2}$
$= \frac{k(6k^{2} - 3k - 1)}{2} + [3(k + 1) - 2]^{2}$
$= \frac{k(6k^{2} - 3k - 1) + 2[3(k + 1) - 2]^{2}}{2}$
$= \frac{k(6k^{2} - 3k - 1) + 2(3k + 1)^{2}}{2}$
$= \frac{k(6k^{2} - 3k - 1) + 18k^{2} + 12k + 2}{2}$
$= \frac{k(6k^{2} - 3k - 1 + 18k + 12) + 2}{2}$
$= \frac{k(6k^{2} + 15k + 11) + 2}{}$
$= \frac{(k + 1)[6k^{2} + 9k + 2]}{2}$
$= \frac{(k + 1)[6(k + 1)^{2} - 3(k + 1) - 1]}{2}$
$= RHS$

Since it is true for n = 1, n = k, and n = k + 1, by the principles of mathematical induction, it is true for all positive values of n.

3 0

3 years ago