Question 969458
Problem 1:


Two positive numbers have a difference of 17. The larger number is one more than twice the smaller. Find the two numbers. 


Step 1: 
write equations given the information.


{{{x-y=17}}}
where x is the first positive number and y is the second positive number.


{{{x=2y+1}}}
x has to be the larger number because the answer is a positive number 
(if x were smaller than y, the answer would be negative, for example: 10-20=-10)


I will break it up into parts


The larger number is one more than twice the smaller.
The larger number: x
is: =
one more than: +1
twice the smaller number: 2y
This combined will give you the equation {{{x=2y+1}}}


Step 2:
solve equations


start with the first equation {{{x-y=17}}}
solve for x
{{{x=17+y}}}


now you can plug in {{{17+y}}} for all the x's in the other equation.



{{{x=2y+1}}}
{{{17+y=2y+1}}}


now you can solve for y


{{{17+y=2y+1}}}
{{{16+y=2y}}}
{{{16=y}}}


The smaller number is 16.


To find the bigger number plug 16 into our first equation


{{{x-y=17}}}
{{{x-16=17}}}
{{{x=33}}}


The 2 positive numbers are 33 and 16.


Problem 2:


A farmer has 162 feet of fence with which to make a corral. If he arranges it into a rectangle that is twice as long as it is wide, what are the dimensions?


Step 1:


Make equations given the information.


The perimeter of a rectangle is {{{P=2L+2W}}}.
we will use this equation
{{{162=2x+2y}}}
where 162 is the length of the fencing available which is also the perimeter of the coral. 
x is the length and y is the width.


a rectangle that is twice as long as it is wide
This can be translated into
{{{x=2y}}}


twice as long as it is wide suggests that the length (x) is going to equal (=) 2 times the width (2y)


Step 2:

solve


now that you have 2 equations you can solve for one of the variables.


{{{x=2y}}}
You can plug 2y into any place in the other equation that has x.


{{{162=2x+2y}}}
{{{162=2(2y)+2y}}}
{{{162=4y+2y}}}
{{{162=6y}}}
{{{27=y}}}


The width is 27.
Now plug in 27.


{{{162=2x+2y}}}
{{{162=2x+2(27)}}}
{{{162=2x+54}}}
{{{108=2x}}}
{{{54=x}}}


The length is 54 and the width is 27.

*notice that the length is twice the width.

you can also check your answer to make sure you didn't make any mistakes in your calculations
{{{162=2x+2y}}}
{{{162=2(54)+2(27)}}}
{{{162=108+54}}}
{{{162=162}}}

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If you have questions, please email me at ericahigley@yahoo.com.