SOLUTION: Recall captain hook's riddle from the beginning of chapter 4: ye have almost found the treasure. start from this spot and walk north and east. three times the sum of the number

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Question 1200212: Recall captain hook's riddle from the beginning of chapter 4:
ye have almost found the treasure. start from this spot and walk north and east. three times the sum of the number of northerly steps and the number of the easterly steps is four more than four times the number of northerly steps. more than this you will need to find the treasure. ye also must know that when ye multiply by five the number two less than the number of northerly steps, ye get the number that is two more than seven times the number of easterly steps.
How many steps north and east should you take to get to the treasure?
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The first task is to translate each verbal sentence into symbolic algebraic equations.

Focus on the first sentence:
"three times the sum of the number of northerly steps and the number of the easterly steps is four more than four times the number of northerly steps."

More specifically, we'll translate the portion highlighted in blue
"three times the sum of the number of northerly steps and the number of the easterly steps is four more than four times the number of northerly steps."

x = number of steps going north
y = number of steps going east
x+y = the sum of those items
3(x+y) = three times that sum
Let's call this expression A. We'll use this later.

Now focus on this portion highlighted in blue
"three times the sum of the number of northerly steps and the number of the easterly steps is four more than four times the number of northerly steps."

x = number of steps going north
4x = four times the number of steps going north
4x+4 = four more compared to the previous quantity
Let's call this expression B.

Expressions A and B are set equal to each other because of the key term "is" in the sentence "three times the sum of the number of northerly steps and the number of the easterly steps is four more than four times the number of northerly steps"
It's like saying "x is 10" which translates to x = 10.

Equating those expressions gives us this equation
3(x+y) = 4x+4



The next sentence to tackle is:
"ye also must know that when ye multiply by five the number two less than the number of northerly steps, ye get the number that is two more than seven times the number of easterly steps."

Let's narrow our focus to the stuff in blue
"ye also must know that when ye multiply by five the number two less than the number of northerly steps, ye get the number that is two more than seven times the number of easterly steps."

x = number of northerly steps
x-2 = two fewer steps compared to before
5(x-2) = five times the previous quantity
Call this expression C.

Change the focus to this portion
"ye also must know that when ye multiply by five the number two less than the number of northerly steps, ye get the number that is two more than seven times the number of easterly steps."

y = number of easterly steps
7y = seven times the previous amount
7y+2 = two more than the previous quantity
Call this expression D.

The expressions C and D are set equal to one another for similar reasoning compared to before.
C = D
5(x-2) = 7y+2

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The key takeaways is what we have this system of equations
system%28%0D%0A3%28x%2By%29+=+4x%2B4%2C%0D%0A5%28x-2%29+=+7y%2B2%0D%0A%29%0D%0A

Let's solve the 2nd equation for x.
5(x-2) = 7y+2
5x-10 = 7y+2
5x = 7y+2+10
5x = 7y+12
x = (7y+12)/5
x = (7y/5)+(12/5)
x = 1.4y + 2.4

Then we'll substitute that into the other equation to solve for y.
3(x+y) = 4x+4
3(1.4y + 2.4+y) = 4(1.4y + 2.4)+4
3(2.4y + 2.4) = 4(1.4y + 2.4)+4
7.2y + 7.2 = 5.6y + 9.6+4
7.2y + 7.2 = 5.6y + 13.6
7.2y-5.6y = 13.6 - 7.2
1.6y = 6.4
y = (6.4)/(1.6)
y = 4

Now we can determine x.
x = 1.4y + 2.4
x = 1.4*4 + 2.4
x = 5.6 + 2.4
x = 8

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Answer:

We found that
x = 8
y = 4

It tells us to go 8 steps north and 4 steps east.