SOLUTION: Using the five-step method solve the following: The sum of a number and its reciprocal is 10/3 (10 on top and 3 on bottom). What is the number

Algebra ->  Equations -> SOLUTION: Using the five-step method solve the following: The sum of a number and its reciprocal is 10/3 (10 on top and 3 on bottom). What is the number      Log On


   

Question 175066: Using the five-step method solve the following:
The sum of a number and its reciprocal is 10/3 (10 on top and 3 on bottom). What is the number
Found 2 solutions by Edwin McCravy, Mathtut:
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Using the five-step method solve the following:
The sum of a number and its reciprocal is 10%2F3. What is the number?


1. Let x = the number

2. Then its reciprocal is 1%2Fx

3. Then the sum of the number and its reciprocal is matrix%281%2C3%2Cx%2C%22%2B%22%2C+1%2Fx%29

4. Then the equation is 

matrix%281%2C5%2Cx%2C%22%2B%22%2C+1%2Fx%2C+%22=%22%2C+10%2F3%29

5. Solve it:

Write the term x as x%2F1 so that all terms
will be fractions.

matrix%281%2C5%2Cx%2F1%2C%22%2B%22%2C+1%2Fx%2C+%22=%22%2C+10%2F3%29

Put parentheses around all the terms:

matrix%281%2C5%2C%28x%2F1%29%2C%22%2B%22%2C+%281%2Fx%29%2C+%22=%22%2C+%2810%2F3%29%29

Clear of fractions.  That is, get the LCD, 
which is 3x, write it as
%28%283x%29%2F1%29, and multiply it by every term:



Cancel what will cancel:



All that's left is this fractionless equation:

matrix%281%2C5%2C3x%5E2%2C+%22%2B%22%2C+3%2C+%22=%22%2C+10x%29

Can you solve that quadratic? If not, post again asking how.

Solutions:  x=1%2F3 and x=3

Edwin

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
I dont know what you 5 step method is(probably in you book) but here is how I would do it
:
we have an unknown number...lets call it a
:
now we must read the word problem and turn it into an equation:
:
sum(meaning we will have addition involved) of the number(a) and its reciprical
(1/a) is(=) 10/3
:
so
a+(1/a)=10/3............now lets start getting rid of those fractions....multiply all terms by 3
:
3a+(3/a)=10.......multiply all terms by a
:
3a%5E2%2B3=10a....now write this in quadradic form
:
3a%5E2-10a%2B3=0 try factoring
:
3a-1%29%28a-3%29
:
system%28a=1%2F3%2Ca=3%29
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-10x%2B3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-10%29%5E2-4%2A3%2A3=64.

Discriminant d=64 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+64+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+64+%29%29%2F2%5C3+=+3
x%5B2%5D+=+%28-%28-10%29-sqrt%28+64+%29%29%2F2%5C3+=+0.333333333333333

Quadratic expression 3x%5E2%2B-10x%2B3 can be factored:
3x%5E2%2B-10x%2B3+=+%28x-3%29%2A%28x-0.333333333333333%29
Again, the answer is: 3, 0.333333333333333. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-10%2Ax%2B3+%29

: