# Graphical solution

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## Introduction

In grade 10, you learnt how to solve sets of simultaneous equations where both equations were linear (i.e. had the highest power equal to 1). In this chapter, you will learn how to solve sets of simultaneous equations where one is linear and one is quadratic. As in Grade 10, the solution will be found both algebraically and graphically.

The only difference between a system of linear simultaneous equations and a system of simultaneous equations with one linear and one quadratic equation, is that the second system will have at most two solutions.

An example of a system of simultaneous equations with one linear equation and one quadratic equation is:

$\begin{array}{c}\hfill y-2x=-4\\ \hfill {x}^{2}+y=4\end{array}$

## Graphical solution

The method of graphically finding the solution to one linear and one quadratic equation is identical to systems of linear simultaneous equations.

## Method: graphical solution to a system of simultaneous equations with one linear and one quadratic equation

1. Make $y$ the subject of each equation.
2. Draw the graphs of each equation as defined above.
3. The solution of the set of simultaneous equations is given by the intersection points of the two graphs.

For this example, making $y$ the subject of each equation, gives:

$\begin{array}{c}\hfill y=2x-4\\ \hfill y=4-{x}^{2}\end{array}$

Plotting the graph of each equation, gives a straight line for the first equation and a parabola for the second equation.

The parabola and the straight line intersect at two points: (2,0) and (-4,-12). Therefore, the solutions to the system of equations in [link] is $x=2,y=0$ and $x=-4,y=12$

Solve graphically:

$\begin{array}{ccc}\hfill y-{x}^{2}+9& =& 0\hfill \\ \hfill y+3x-9& =& 0\hfill \end{array}$
1. For the first equation:

$\begin{array}{ccc}\hfill y-{x}^{2}+9& =& 0\hfill \\ \hfill y& =& {x}^{2}-9\hfill \end{array}$

and for the second equation:

$\begin{array}{ccc}\hfill y+3x-9& =& 0\hfill \\ \hfill y& =& -3x+9\hfill \end{array}$
2. The graphs intersect at $\left(-6,27\right)$ and at $\left(3,0\right)$ .

3. The first solution is $x=-6$ and $y=27$ . The second solution is $x=3$ and $y=0$ .

## Graphical solution

Solve the following systems of equations graphically. Leave your answer in surd form, where appropriate.

1. ${b}^{2}-1-a=0,a+b-5=0$
2. $x+y-10=0,{x}^{2}-2-y=0$
3. $6-4x-y=0,12-2{x}^{2}-y=0$
4. $x+2y-14=0,{x}^{2}+2-y=0$
5. $2x+1-y=0,25-3x-{x}^{2}-y=0$

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Other chapter Q/A we can ask