Question 855172: 1. Divide 8 into two parts such that the sum of their squares is 34.
2. $9000 were divided equally among a certain number of persons. Had there been 20 persons more, each would have got $160 less. Find the original number of persons.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = one of the parts.
let y = the other of the parts.
you get:
x*y = 8 ***** first equation
x^2 + y^2 = 34 ***** second equation
solve for x in the first equation to get:
x = 8/y
substitute for x in the second equation to get:
(8/y)^2 + y^2 = 34
simplify this equation to get:
64/y^2 + y^2 = 34
multiply both sides of this equation by y^2 to get:
64 + y^4 = 34y^2
reorder the terms to get:
y^4 + 64 = 34y^2
subtract 34y^2 from both sides of this equation to get:
y^4 - 34y^2 + 64 = 0
factor this equation to get:
(y^2 - 32) * (y^2 - 2) = 0
solve for y^2 in each of the factors to get:
y^2 = 32
y^2 = 2
solve for y in each of these equations to get:
y = + sqrt(32)
y = - sqrt(32)
y = + sqrt(2)
y = - sqrt(2)
for each of these values of y, solve for x using the first equation.
first equation is:
x * y = 8
solve for x in this equation to get:
x = 8/y
use the values of y you just solved for to get the (x,y) pairs that will be the possible solutions to this problem.
you get:
when y = + sqrt(32), x = + 8/sqrt(32)
when y = - sqrt(32), x = - 8/sqrt(32)
when y = + sqrt(2), x = + 8/sqrt(2)
when y = - sqrt(2), x = - 8/sqrt(2)
those are your possible solutions.
you can confirm by multiplying x * y to make sure you get 8.
you can also confirm by adding x^2 + y^2 to make sure you get 34.
i'll do one of the possible solutions to show you.
you can do the rest.
when x = 8/sqrt(32) and y = sqrt(32), you get:
x * y = sqrt(32) * 8/sqrt(32) which is equal to 8.
x^2 + y^2 = (8/sqrt(32))^2 + sqrt(32)^2 which is equal to:
8^2 / sqrt(32)^2 + sqrt(32)^2 which is equal to:
64 / 32 + 32 which is equal to:
2 + 32 which is equal to 34.
you can also graph your original equations to find the intesection points.
those intersection points will agree with the solution just shown above.
that graph looks like:
sqrt(32) is equal to approximately 5.7
sqrt(2) is equal to approximately 1.4
8/sqrt(32) is equal to approximately 1.4
8/sqrt(2) is equal to approximately 5.7
note that sqrt(32) is the same as 8/sqrt(2) and sqrt(2) is the same as 8/sqrt(32).
this could throw you off but that's perfectly legitimate in this case.
i'll show you one of them and you can prove the other one to yourself.
i'll show you why sqrt(32) is the same as 8 / sqrt(2)
start with sqrt(32) = 8/sqrt(2)
square both sides of this equation to get:
32 = 64 / 2
simplify to get:
32 = 32
this confirms they are equivalent.
another way to confirm it is:
start with sqrt(32) = 8/sqrt(2)
multiply both sides of this equation by sqrt(2) to get:
sqrt(32) * sqrt(2) = 8
since sqrt(a) * sqrt(b) = sqrt(a*b), this equation becomes:
sqrt(32*2) = 8
simplify to get:
sqrt(64) = 8
simplify further to get:
8 = 8
the equality is confirmed again, only using a different method, both of which follow the rules of mathematical operations.
your solutions to this problem are:
when y = + sqrt(32), x = + 8/sqrt(32)
when y = - sqrt(32), x = - 8/sqrt(32)
when y = + sqrt(2), x = + 8/sqrt(2)
when y = - sqrt(2), x = - 8/sqrt(2)
now we'll look at your second question.
that question is:
2. $9000 were divided equally among a certain number of persons. Had there been 20 persons more, each would have got $160 less. Find the original number of persons.
let x = number of people
let y = amount each person received originally
you get 2 equations.
y = 9000/x ***** first equation
(y-160) = 9000 / (x+20) ***** second equation
add 160 to both sides of the second equation to get:
y = 9000 / (x+20) + 160
since 9000/x and 9000 / (x+20) + 160 are both equal to y, then they are both equal to each other, so you get:
9000/x = 9000/(x+20) + 160
multiply both sides of this equation by x * (x+20) to get:
9000 * (x+20) = (9000 * x) + (160 * x * (x+20))
simplify this equation to get:
9000*x + 9000*20 = 9000*x + 160*(x^2+20x)
simplify further to get:
9000*x + 180000 = 9000*x + 160*x^2 + 3200*x
subtract 9000*x and 180000 from both sides of the equation to get:
0 = 9000*x - 9000*x + 160*x^2 + 3200*x - 180000
simplify to get:
0 = 160*x^2 + 3200*x - 180000
divide both sides of this equation by 160 to get:
0 = x^2 + 20x - 1125
since, if a = b, then b = a, this is the same as:
x^2 + 20x - 1125 = 0
factor this equation to get:
(x + 45) * (x - 25) = 0
solve for x to get:
x = -45
x = 25
since the number of people can't be negative, then your only solution is:
x = 25
solve for y in the equation of y = 9000/x to get:
y = 9000/25 = 360
your solution appears to be:
x = 25
y = 360
confirm by substituting these values in the original equations.
y = 9000/x becomes 360 = 9000/25 which becomes 360 = 360.
solution is confirmed for first equation.
y-160 = 9000 / (x+20) becomes 360 - 160 = 9000 / 45 which becomes 200 = 200.
solution is confirmed for second equation.
your solution is:
x = 25 which means that the original number of people was 25.
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