# 5.7 Linear equations in two variables  (Page 2/2)

 Page 2 / 2

These are only three of the infintely many solutions to this equation.

## Sample set a

Find a solution to each of the following linear equations in two variables and write the solution as an ordered pair.

$y=3x-6,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=1$

Substitute 1 for $x$ , compute, and solve for $y$ .

$\begin{array}{l}y=3\left(1\right)-6\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=3-6\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-3\end{array}$

Hence, one solution is $\left(1,\text{\hspace{0.17em}}-3\right)$ .

$y=15-4x,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-10$

Substitute $-10$ for $x$ , compute, and solve for $y$ .

$\begin{array}{l}y=15-4\left(-10\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=15+40\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=55\end{array}$

Hence, one solution is $\left(-10,\text{\hspace{0.17em}}55\right)$ .

$b=-9a+21,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}a=2$

Substitute 2 for $a$ , compute, and solve for $b$ .

$\begin{array}{l}b=-9\left(2\right)+21\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-18+21\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=3\end{array}$

Hence, one solution is $\left(2,\text{\hspace{0.17em}}3\right)$ .

$5x-2y=1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=0$

Substitute 0 for $x$ , compute, and solve for $y$ .

$\begin{array}{rrr}\hfill 5\left(0\right)-2y& \hfill =& \hfill 1\\ \hfill 0-2y& \hfill =& \hfill 1\\ \hfill -2y& \hfill =& \hfill 1\\ \hfill y& \hfill =& \hfill -\frac{1}{2}\end{array}$

Hence, one solution is $\left(0,\text{\hspace{0.17em}}-\frac{1}{2}\right)$ .

## Practice set a

Find a solution to each of the following linear equations in two variables and write the solution as an ordered pair.

$y=7x-20,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=3$

$\left(3,\text{\hspace{0.17em}}1\right)$

$m=-6n+1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}n=2$

$\left(2,\text{\hspace{0.17em}}-11\right)$

$b=3a-7,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}a=0$

$\left(0,\text{\hspace{0.17em}}-7\right)$

$10x-5y-20=0,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-8$

$\left(-8,\text{\hspace{0.17em}}-20\right)$

$3a+2b+6=0,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}a=-1$

$\left(-1,\text{\hspace{0.17em}}\frac{-3}{2}\right)$

## Exercises

For the following problems, solve the linear equations in two variables.

$y=8x+14,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=1$

$\left(1,22\right)$

$y=-2x+1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=0$

$y=5x+6,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=4$

$\left(4,26\right)$

$x+y=7,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=8$

$3x+4y=0,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-3$

$\left(-3,\frac{9}{4}\right)$

$-2x+y=1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=\frac{1}{2}$

$5x-3y+1=0,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-6$

$\left(-6,-\frac{29}{3}\right)$

$-4x-4y=4,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}y=7$

$2x+6y=1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}y=0$

$\left(\frac{1}{2},0\right)$

$-x-y=0,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}y=\frac{14}{3}$

$y=x,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=1$

$\left(1,1\right)$

$x+y=0,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=0$

$y+\frac{3}{4}=x,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=\frac{9}{4}$

$\left(\frac{9}{4},\frac{3}{2}\right)$

$y+17=x,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-12$

$-20y+14x=1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=8$

$\left(8,\frac{111}{20}\right)$

$\frac{3}{5}y+\frac{1}{4}x=\frac{1}{2},\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-3$

$\frac{1}{5}x+y=-9,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}y=-1$

$\left(-40,-1\right)$

$y+7-x=0,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=$

$2x+31y-3=0,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=a$

$\left(a,\frac{3-2a}{31}\right)$

$436x+189y=881,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-4231$

$y=6\left(x-7\right),\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=2$

$\left(2,-30\right)$

$y=2\left(4x+5\right),\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-1$

$5y=9\left(x-3\right),\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=2$

$\left(2,-\frac{9}{5}\right)$

$3y=4\left(4x+1\right),\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-3$

$-2y=3\left(2x-5\right),\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=6$

$\left(6,-\frac{21}{2}\right)$

$-8y=7\left(8x+2\right),\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=0$

$b=4a-12,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}a=-7$

$\left(-7,-40\right)$

$b=-5a+21,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}a=-9$

$4b-6=2a+1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}a=0$

$\left(0,\frac{7}{4}\right)$

$-5m+11=n+1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}n=4$

$3\left(t+2\right)=4\left(s-9\right),\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}s=1$

$\left(1,-\frac{38}{3}\right)$

$7\left(t-6\right)=10\left(2-s\right),\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}s=5$

$y=0x+5,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=1$

$\left(1,5\right)$

$2y=0x-11,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=-7$

$-y=0x+10,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=3$

$\left(3,-10\right)$

$-5y=0x-1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=0$

$y=0\left(x-1\right)+6,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=1$

$\left(1,6\right)$

$y=0\left(3x+9\right)-1,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x=12$

## Calculator problems

An examination of the winning speeds in the Indianapolis 500 automobile race from 1961 to 1970 produces the equation $y=1.93x+137.60$ , where $x$ is the number of years from 1960 and $y$ is the winning speed. Statistical methods were used to obtain the equation, and, for a given year, the equation gives only the approximate winning speed. Use the equation $y=1.93x+137.60$ to find the approximate winning speed in

1. 1965
2. 1970
3. 1986
4. 1990

(a) Approximately 147 mph using $\left(5,147.25\right)$
(b) Approximately 157 mph using $\left(10,156.9\right)$
(c) Approximately 188 mph using $\left(26,187.78\right)$
(d) Approximately 196 mph using $\left(30,195.5\right)$

In electricity theory, Ohm’s law relates electrical current to voltage by the equation $y=0.00082x$ , where $x$ is the voltage in volts and $y$ is the current in amperes. This equation was found by statistical methods and for a given voltage yields only an approximate value for the current. Use the equation $y=0.00082x$ to find the approximate current for a voltage of

1. 6 volts
2. 10 volts

Statistical methods have been used to obtain a relationship between the actual and reported number of German submarines sunk each month by the U.S. Navy in World War II. The equation expressing the approximate number of actual sinkings, $y$ , for a given number of reported sinkings, $x$ , is $y=1.04x+0.76$ . Find the approximate number of actual sinkings of German submarines if the reported number of sinkings is

1. 4
2. 9
3. 10

(a) Approximately 5 sinkings using $\left(4,4.92\right)$
(b) Approximately 10 sinkings using $\left(9,10.12\right)$
(c) Approximately 11 sinkings using $\left(10,11.16\right)$

Statistical methods have been used to obtain a relationship between the heart weight (in milligrams) and the body weight (in milligrams) of 10-month-old diabetic offspring of crossbred male mice. The equation expressing the approximate body weight for a given heart weight is $y=0.213x-4.44$ . Find the approximate body weight for a heart weight of

1. 210 mg
2. 245 mg

Statistical methods have been used to produce the equation $y=0.176x-0.64$ . This equation gives the approximate red blood cell count (in millions) of a dog’s blood, $y$ , for a given packed cell volume (in millimeters), $x$ . Find the approximate red blood cell count for a packed cell volume of

1. 40 mm
2. 42 mm

(a) Approximately $6.4$ using $\left(40,6.4\right)$
(b) Approximately $4.752$ using $\left(42,7.752\right)$

An industrial machine can run at different speeds. The machine also produces defective items, and the number of defective items it produces appears to be related to the speed at which the machine is running. Statistical methods found that the equation $y=0.73x-0.86$ is able to give the approximate number of defective items, $y$ , for a given machine speed, $x$ . Use this equation to find the approximate number of defective items for a machine speed of

1. 9
2. 12

A computer company has found, using statistical techniques, that there is a relationship between the aptitude test scores of assembly line workers and their productivity. Using data accumulated over a period of time, the equation $y=0.89x-41.78$ was derived. The $x$ represents an aptitude test score and $y$ the approximate corresponding number of items assembled per hour. Estimate the number of items produced by a worker with an aptitude score of

1. 80
2. 95

(a) Approximately 29 items using $\left(80,29.42\right)$
(b) Approximately 43 items using $\left(95,42.77\right)$

Chemists, making use of statistical techniques, have been able to express the approximate weight of potassium bromide, $W$ , that will dissolve in 100 grams of water at $T$ degrees centigrade. The equation expressing this relationship is $W=0.52T+54.2$ . Use this equation to predict the potassium bromide weight that will dissolve in 100 grams of water that is heated to a temperature of

1. 70 degrees centigrade
2. 95 degrees centigrade

The marketing department at a large company has been able to express the relationship between the demand for a product and its price by using statistical techniques. The department found, by analyzing studies done in six different market areas, that the equation giving the approximate demand for a product (in thousands of units) for a particular price (in cents) is $y=-14.15x+257.11$ . Find the approximate number of units demanded when the price is

1. $0.12$
2. $0.15$

(a) Approximately 87 units using $\left(12,87.31\right)$
(b) Approximately 45 units using $\left(15,44.86\right)$

The management of a speed-reading program claims that the approximate speed gain (in words per minute), $G$ , is related to the number of weeks spent in its program, $W$ , is given by the equation $G=26.68W-7.44$ . Predict the approximate speed gain for a student who has spent

1. 3 weeks in the program
2. 10 weeks in the program

## Exercises for review

( [link] ) Find the product. $\left(4x-1\right)\left(3x+5\right)$ .

$12{x}^{2}+17x-5$

( [link] ) Find the product. $\left(5x+2\right)\left(5x-2\right)$ .

( [link] ) Solve the equation $6\left[2\left(x-4\right)+1\right]=3\left[2\left(x-7\right)\right]$ .

$x=0$

( [link] ) Solve the inequality $-3a-\left(a-5\right)\ge a+10$ .

( [link] ) Solve the compound inequality $-1<4y+11<27$ .

$-3

#### Questions & Answers

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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