SOLUTION: I don't understand how to arrive at the answers to the following problems: A)One number is 2 more than 5 times another. Their product is 24. Find the numbers. B) One number i

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Question 352819: I don't understand how to arrive at the answers to the following problems:
A)One number is 2 more than 5 times another. Their product is 24. Find the numbers.
B) One number is 4 times another. Their product is 4 times their sum. Find the numbers.
Thank you!
Found 2 solutions by jim_thompson5910, jrfrunner:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started

# 1

"One number is 2 more than 5 times another" means that y=5x%2B2 and "Their product is 24" tells us that xy=24 (x times y = 24)


xy=24 Start with the second equation.


x%285x%2B2%29=24 Plug in y=5x%2B2


5x%5E2%2B2x=24 Distribute.


5x%5E2%2B2x-24=0 Subtract 24 from both sides.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=5, b=2, and c=-24


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%282%29+%2B-+sqrt%28+%282%29%5E2-4%285%29%28-24%29+%29%29%2F%282%285%29%29 Plug in a=5, b=2, and c=-24


x+=+%28-2+%2B-+sqrt%28+4-4%285%29%28-24%29+%29%29%2F%282%285%29%29 Square 2 to get 4.


x+=+%28-2+%2B-+sqrt%28+4--480+%29%29%2F%282%285%29%29 Multiply 4%285%29%28-24%29 to get -480


x+=+%28-2+%2B-+sqrt%28+4%2B480+%29%29%2F%282%285%29%29 Rewrite sqrt%284--480%29 as sqrt%284%2B480%29


x+=+%28-2+%2B-+sqrt%28+484+%29%29%2F%282%285%29%29 Add 4 to 480 to get 484


x+=+%28-2+%2B-+sqrt%28+484+%29%29%2F%2810%29 Multiply 2 and 5 to get 10.


x+=+%28-2+%2B-+22%29%2F%2810%29 Take the square root of 484 to get 22.


x+=+%28-2+%2B+22%29%2F%2810%29 or x+=+%28-2+-+22%29%2F%2810%29 Break up the expression.


x+=+%2820%29%2F%2810%29 or x+=++%28-24%29%2F%2810%29 Combine like terms.


x+=+2 or x+=+-12%2F5 Simplify.


So the possible answers are x+=+2 or x+=+-12%2F5


If you want integer only answers, then the only solution is x=2


So one number is 2 and the other is y=5%282%29%2B2=10%2B2=12


So the two numbers are 2 and 12.

Answer by jrfrunner(365) About Me  (Show Source):
You can put this solution on YOUR website!
let x=one number
let y=the other number
--
you are given: One number is 2 more than 5 times another
x=2+5y
--
also you are told: Their product is 24
x*y=24
==
substitute x=2+5y into x*y=24 in order to reduce the problem by one variable
x*y=(2+5y)*y=24
--
2y%2B5y%5E2=24+
5y%5E2%2B2y-24=0
--
use the quadratic formula to solve for y
where a=5, b=2 and c=-24
--
y=%28-2-Sqrt%28%28-2%29%5E2-4%2A5%2A%28-24%29%29%29%2F%282%2A5%29 or y=%28-2%2BSqrt%28%28-2%29%5E2-4%2A5%2A%28-24%29%29%29%2F%282%2A5%29
--
y=(-2-22)/10=-2.4 or y=(-2+22)/10=2
==
substitute y=-2.4 or y=2 into x=2+5y to find x
x=2+5(-2.4)=-10 and x=2+5*(2)=12
--
so the possible answers are x=-10 and y=-2.4 or x=12 and y=2