# 11.3 Elimination by addition  (Page 2/2)

 Page 2 / 2

## Practice set a

$\left\{\begin{array}{l}x+y=6\\ 2x-y=0\end{array}$

$\left(2,4\right)$

$\left\{\begin{array}{l}x+6y=8\\ -x-2y=0\end{array}$

$\left(-4,2\right)$

## Sample set b

Solve the following systems using the addition method.

Solve $\left\{\begin{array}{rrr}\hfill 6a-5b=14& \hfill & \hfill \left(1\right)\\ \hfill 2a+2b=-10& \hfill & \hfill \left(2\right)\end{array}$

Step 1: The equations are already in the proper form, $ax+by=c.$

Step 2: If we multiply equation (2) by —3, the coefficients of $a$ will be opposites and become 0 upon addition, thus eliminating $a$ .

$\begin{array}{lllll}\left\{\begin{array}{l}6a-5b=14\\ -3\left(2a+2b\right)=-3\left(10\right)\end{array}\hfill & \hfill & \to \hfill & \hfill & \left\{\begin{array}{l}6a-5b=14\\ -6a-6b=30\end{array}\hfill \end{array}$

$\frac{\begin{array}{c}6a-5b=14\\ -6a-6b=30\end{array}}{0-11b=44}$

Step 4:  Solve the equation $-11b=44.$

$-11b=44$
$b=-4$

Step 5:  Substitute $b=-4$ into either of the original equations. We will use equation 2.

$\begin{array}{rrrrr}\hfill 2a+2b& \hfill =& \hfill -10& \hfill & \hfill \\ \hfill 2a+2\left(-4\right)& \hfill =& \hfill -10& \hfill & \hfill \text{Solve\hspace{0.17em}for\hspace{0.17em}}a.\\ \hfill 2a-8& \hfill =& \hfill -10& \hfill & \hfill \\ \hfill 2a& \hfill =& \hfill -2& \hfill & \hfill \\ \hfill a& \hfill =& \hfill -1& \hfill & \hfill \end{array}$

We now have $a=-1$ and $b=-4.$

Step 6:  Substitute $a=-1$ and $b=-4$ into both the original equations for a check.

$\begin{array}{rrrrrrrrrrrr}\hfill \left(1\right)& \hfill & \hfill 6a-5b& \hfill =& \hfill 14& \hfill & \hfill \left(2\right)& \hfill & \hfill 2a+2b& \hfill =& \hfill -10& \hfill \\ \hfill & \hfill & \hfill 6\left(-1\right)-5\left(-4\right)& \hfill =& \hfill 14& \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill & \hfill 2\left(-1\right)+2\left(-4\right)& \hfill =& \hfill -10& \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill -6+20& \hfill =& \hfill 14& \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill & \hfill -2-8& \hfill =& \hfill -10& \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 14& \hfill =& \hfill 14& \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill & \hfill & \hfill -10& \hfill =& \hfill -10& \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}$

Step 7:  The solution is $\left(-1,-4\right).$

Solve  $\left\{\begin{array}{ccc}\begin{array}{l}3x+2y=-4\hfill \\ 4x=5y+10\hfill \end{array}& & \begin{array}{l}\left(1\right)\hfill \\ \left(2\right)\hfill \end{array}\end{array}$

Step 1:  Rewrite the system in the proper form.

$\left\{\begin{array}{ccc}\begin{array}{l}3x+2y=-4\\ 4x-5y=10\end{array}& & \begin{array}{l}\left(1\right)\\ \left(2\right)\end{array}\end{array}$

Step 2:  Since the coefficients of $y$ already have opposite signs, we will eliminate $y$ .
Multiply equation (1) by 5, the coefficient of $y$ in equation 2.
Multiply equation (2) by 2, the coefficient of $y$ in equation 1.

$\begin{array}{rrrrr}\hfill \left\{\begin{array}{l}5\left(3x+2y\right)=5\left(-4\right)\\ 2\left(4x-5y\right)=2\left(10\right)\end{array}& \hfill & \hfill \to & \hfill & \hfill \left\{\begin{array}{l}15x+10y=-20\\ 8x-10y=20\end{array}\end{array}$

$\frac{\begin{array}{c}15x+10y=-20\\ 8x-10y=20\end{array}}{23x+0=0}$

Step 4:  Solve the equation $23x=0$

$23x=0$

$x=0$

Step 5:  Substitute $x=0$ into either of the original equations. We will use equation 1.

$\begin{array}{rrrrr}\hfill 3x+2y& \hfill =& \hfill -4& \hfill & \hfill \\ \hfill 3\left(0\right)+2y& \hfill =& \hfill -4& \hfill & \hfill \text{Solve\hspace{0.17em}for\hspace{0.17em}}y.\\ \hfill 0+2y& \hfill =& \hfill -4& \hfill & \hfill \\ \hfill y& \hfill =& \hfill -2& \hfill & \hfill \end{array}$

We now have $x=0$ and $y=-2.$

Step 6:  Substitution will show that these values check.

Step 7:  The solution is $\left(0,-2\right).$

## Practice set b

Solve each of the following systems using the addition method.

$\left\{\begin{array}{l}3x+y=1\\ 5x+y=3\end{array}$

$\left(1,-2\right)$

$\left\{\begin{array}{l}x+4y=1\\ x-2y=-5\end{array}$

$\left(-3,1\right)$

$\left\{\begin{array}{l}2x+3y=-10\\ -x+2y=-2\end{array}$

$\left(-2,-2\right)$

$\left\{\begin{array}{l}5x-3y=1\\ 8x-6y=4\end{array}$

$\left(-1,-2\right)$

$\left\{\begin{array}{l}3x-5y=9\\ 4x+8y=12\end{array}$

$\left(3,0\right)$

## Addition and parallel or coincident lines

When the lines of a system are parallel or coincident, the method of elimination produces results identical to that of the method of elimination by substitution.

If computations eliminate all variables and produce a contradiction, the two lines of the system are parallel and the system is called inconsistent.

If computations eliminate all variables and produce an identity, the two lines of the system are coincident and the system is called dependent.

## Sample set c

Solve $\left\{\begin{array}{rrr}\hfill 2x-y=1& \hfill & \hfill \left(1\right)\\ \hfill 4x-2y=4& \hfill & \hfill \left(2\right)\end{array}$

Step 1: The equations are in the proper form.

Step 2: We can eliminate $x$ by multiplying equation (1) by –2.

$\begin{array}{rrrrr}\hfill \left\{\begin{array}{c}-2\left(2x-y\right)=-2\left(1\right)\\ 4x-2y=4\end{array}& \hfill & \hfill \to & \hfill & \hfill \left\{\begin{array}{c}-4x+2y=-2\\ 4x-2y=4\end{array}\end{array}$

$\frac{\begin{array}{c}-4x+2y=-2\\ 4x-2y=4\end{array}}{\begin{array}{c}0+0=2\\ 0=2\end{array}}$

This is false and is therefore a contradiction. The lines of this system are parallel.  This system is inconsistent.

Solve  $\left\{\begin{array}{rrr}\hfill 4x+8y=8& \hfill & \hfill \left(1\right)\\ \hfill 3x+6y=6& \hfill & \hfill \left(2\right)\end{array}$

Step 1:  The equations are in the proper form.

Step 2:  We can eliminate $x$ by multiplying equation (1) by –3 and equation (2) by 4.

$\begin{array}{rrrrr}\hfill \left\{\begin{array}{c}-3\left(4x+8y\right)=-3\left(8\right)\\ 4\left(3x+6y\right)=4\left(6\right)\end{array}& \hfill & \hfill \to & \hfill & \hfill \left\{\begin{array}{c}-12x-24y=-24\\ 12x+24y=24\end{array}\end{array}$

$\frac{\begin{array}{c}-12x-24y=-24\\ 12x+24y=24\end{array}}{\begin{array}{c}0+0=0\\ 0=0\end{array}}$

This is true and is an identity. The lines of this system are coincident.

This system is dependent.

## Practice set c

Solve each of the following systems using the addition method.

$\left\{\begin{array}{c}-x+2y=6\\ -6x+12y=1\end{array}$

inconsistent

$\left\{\begin{array}{c}4x-28y=-4\\ x-7y=-1\end{array}$

dependent

## Exercises

For the following problems, solve the systems using elimination by addition.

$\left\{\begin{array}{c}x+y=11\\ x-y=-1\end{array}$

$\left(5,6\right)$

$\left\{\begin{array}{c}x+3y=13\\ x-3y=-11\end{array}$

$\left\{\begin{array}{c}3x-5y=-4\\ -4x+5y=2\end{array}$

$\left(2,2\right)$

$\left\{\begin{array}{c}2x-7y=1\\ 5x+7y=-22\end{array}$

$\left\{\begin{array}{c}-3x+4y=-24\\ 3x-7y=42\end{array}$

$\left(0,-6\right)$

$\left\{\begin{array}{c}8x+5y=3\\ 9x-5y=-71\end{array}$

$\left\{\begin{array}{c}-x+2y=-6\\ x+3y=-4\end{array}$

$\left(2,-2\right)$

$\left\{\begin{array}{c}4x+y=0\\ 3x+y=0\end{array}$

$\left\{\begin{array}{c}x+y=-4\\ -x-y=4\end{array}$

dependent

$\left\{\begin{array}{c}-2x-3y=-6\\ 2x+3y=6\end{array}$

$\left\{\begin{array}{c}3x+4y=7\\ x+5y=6\end{array}$

$\left(1,1\right)$

$\left\{\begin{array}{c}4x-2y=2\\ 7x+4y=26\end{array}$

$\left\{\begin{array}{c}3x+y=-4\\ 5x-2y=-14\end{array}$

$\left(-2,2\right)$

$\left\{\begin{array}{c}5x-3y=20\\ -x+6y=-4\end{array}$

$\left\{\begin{array}{c}6x+2y=-18\\ -x+5y=19\end{array}$

$\left(-4,3\right)$

$\left\{\begin{array}{c}x-11y=17\\ 2x-22y=4\end{array}$

$\left\{\begin{array}{c}-2x+3y=20\\ -3x+2y=15\end{array}$

$\left(-1,6\right)$

$\left\{\begin{array}{c}-5x+2y=-4\\ -3x-5y=10\end{array}$

$\left\{\begin{array}{c}-3x-4y=2\\ -9x-12y=6\end{array}$

dependent

$\left\{\begin{array}{c}3x-5y=28\\ -4x-2y=-20\end{array}$

$\left\{\begin{array}{c}6x-3y=3\\ 10x-7y=3\end{array}$

$\left(1,\text{\hspace{0.17em}}1\right)$

$\left\{\begin{array}{c}-4x+12y=0\\ -8x+16y=0\end{array}$

$\left\{\begin{array}{c}3x+y=-1\\ 12x+4y=6\end{array}$

inconsistent

$\left\{\begin{array}{c}8x+5y=-23\\ -3x-3y=12\end{array}$

$\left\{\begin{array}{c}2x+8y=10\\ 3x+12y=15\end{array}$

dependent

$\left\{\begin{array}{c}4x+6y=8\\ 6x+8y=12\end{array}$

$\left\{\begin{array}{c}10x+2y=2\\ -15x-3y=3\end{array}$

inconsistent

$\left\{\begin{array}{c}x+\frac{3}{4}y=-\frac{1}{2}\\ \frac{3}{5}x+y=-\frac{7}{5}\end{array}$

$\left\{\begin{array}{c}x+\frac{1}{3}y=\frac{4}{3}\\ -x+\frac{1}{6}y=\frac{2}{3}\end{array}$

$\left(0,4\right)$

$\left\{\begin{array}{c}8x-3y=25\\ 4x-5y=-5\end{array}$

$\left\{\begin{array}{c}-10x-4y=72\\ 9x+5y=39\end{array}$

$\left(-\frac{258}{7},\frac{519}{7}\right)$

$\left\{\begin{array}{c}12x+16y=-36\\ -10x+12y=30\end{array}$

$\left\{\begin{array}{c}25x-32y=14\\ -50x+64y=-28\end{array}$

dependent

## Exercises for review

( [link] ) Simplify and write ${\left(2{x}^{-3}{y}^{4}\right)}^{5}{\left(2x{y}^{-6}\right)}^{-5}$ so that only positive exponents appear.

( [link] ) Simplify $\sqrt{8}+3\sqrt{50}.$

$17\sqrt{2}$

( [link] ) Solve the radical equation $\sqrt{2x+3}+5=8.$

( [link] ) Solve by graphing $\left\{\begin{array}{c}x+y=4\\ 3x-y=0\end{array}$

$\left(1,3\right)$

( [link] ) Solve using the substitution method: $\left\{\begin{array}{c}3x-4y=-11\\ 5x+y=-3\end{array}$

Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!