# 3.6 Using letters as numbers

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This module shows how it can be helpful, on occasion, to use letters as numbers in order to quickly find solutions to variations on a simultaneous equation problem.

Toward the end of this chapter, there are some problems in substitution and elimination where letters are used in place of numbers . For instance, consider the following problem:

$2y-\text{ax}=7$
$4y+3\text{ax}=9$

What do we do with those "a"s? Like any other variable, they simply represent an unknown number. As we solve for $x$ , we will simply leave $a$ as a variable.

This problem lends itself more naturally to elimination than to substitution, so I will double the top equation and then subtract the two equations and solve.

$\begin{array}{c}4y-2\text{ax}=\text{14}\\ \underline{-\left(4y+3\text{ax}=9\right)}\\ 0y-5\text{ax}=5\end{array}$
$x=\frac{5}{-5a}=\frac{-1}{a}$

As always, we can solve for the second variable by plugging into either of our original equations.

$2y-a\left(\frac{-1}{a}\right)=7$
$2y+1=7$
$y=3$

There is no new math here, just elimination. The real trick is not to be spooked by the $a$ , and do the math just like you did before.

And what does that mean? It means we have found a solution that works for those two equations, regardless of a. We can now solve the following three problems (and an infinite number of others) without going through the hard work.

If $a=5$ , If $a=10$ , If $a=-3$ ,
The original equations become: The original equations become: The original equations become:
$\begin{array}{c}2y-5x=7\\ 4y+\text{15}x=9\end{array}$ $\begin{array}{c}2y-\text{10}x=7\\ 4y+\text{30}x=9\end{array}$ $\begin{array}{c}2y+3x=7\\ 4y-9x=9\end{array}$
And the solution is: And the solution is: And the solution is:
$x=\frac{-1}{5},y=3$ $x=\frac{-1}{\text{10}},y=3$ $x=\frac{1}{3},y=3$

The whole point is that I did not have to solve those three problems—by elimination, substitution, or anything else. All I had to do was plug $a$ into the general answer I had already found previously. If I had to solve a hundred such problems, I would have saved myself a great deal of time by going through the hard work once to find a general solution!

Mathematicians use this trick all the time. If they are faced with many similar problems, they will attempt to find a general problem that encompasses all the specific problems, by using variables to replace the numbers that change. You will do this in an even more general way in the text, when you solve the “general” simultaneous equations where all the numbers are variables. Then you will have a formula that you can plug any pair of simultaneous equations into to find the answer at once. This formula would also make it very easy, for instance, to program a computer to solve simultaneous equations (computers are terrible at figuring things out, but they’re great at formulas).

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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