5.7 Linear equations in two variables

 Page 1 / 2
<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para>This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules (<link document="m21980"/>) and (<link document="m21979"/>)). Objectives of this module: be able to identify the solution of a linear equation in two variables, know that solutions to linear equations in two variables can be written as ordered pairs.

Overview

• Solutions to Linear Equations in Two Variables
• Ordered Pairs as Solutions

Solution to an equation in two variables

We have discovered that an equation is a mathematical way of expressing the relationship of equality between quantities. If the relationship is between two quantities, the equation will contain two variables. We say that an equation in two variables has a solution if an ordered pair of values can be found such that when these two values are substituted into the equation a true statement results. This is illustrated when we observe some solutions to the equation $y=2x+5$ .

1. $x=4,\text{\hspace{0.17em}}y=13;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{since}\text{\hspace{0.17em}}13=2\left(4\right)+5\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{true}\text{}$ .
2. $x=1,\text{\hspace{0.17em}}y=7;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{since}\text{\hspace{0.17em}}7=2\left(1\right)+5\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{true}\text{}$ .
3. $x=0,\text{\hspace{0.17em}}y=5;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{since}\text{\hspace{0.17em}}5=2\left(0\right)+5\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{true}\text{}$ .
4. $x=-6,\text{\hspace{0.17em}}y=-7;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{since}\text{\hspace{0.17em}}-7=2\left(-6\right)+5\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{true}\text{}$ .

Ordered pairs as solutions

It is important to keep in mind that a solution to a linear equation in two variables is an ordered pair of values, one value for each variable. A solution is not completely known until the values of both variables are specified.

Independent and dependent variables

Recall that, in an equation, any variable whose value can be freely assigned is said to be an independent variable. Any variable whose value is determined once the other value or values have been assigned is said to be a dependent variable. If, in a linear equation, the independent variable is $x$ and the dependent variable is $y$ , and a solution to the equation is $x=a$ and $y=b$ , the solution is written as the

ORDERED PAIR      $\left(a,\text{\hspace{0.17em}}b\right)$

Ordered pair

In an ordered pair , $\left(a,\text{\hspace{0.17em}}b\right)$ , the first component, $a$ , gives the value of the independent variable, and the second component, $b$ , gives the value of the dependent variable.

We can use ordered pairs to show some solutions to the equation $y=6x-7$ .

$\left(0,-7\right)$ .
If $x=0$ and $y=-7$ , we get a true statement upon substitution and computataion.

$\begin{array}{lllll}\hfill y& =\hfill & 6x-7\hfill & \hfill & \hfill \\ -7\hfill & =\hfill & 6\left(0\right)-7\hfill & \hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ -7\hfill & =\hfill & -7\hfill & \text{}\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

$\left(8,\text{\hspace{0.17em}}41\right)$ .
If $x=8$ and $y=41$ , we get a true statement upon substitution and computataion.

$\begin{array}{lllll}y\hfill & =\hfill & 6x-7\hfill & \hfill & \hfill \\ 41\hfill & =\hfill & 6\left(8\right)-7\hfill & \hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ 41\hfill & =\hfill & 48-7\hfill & \hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ 41\hfill & =\hfill & 41\hfill & \text{}\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

$\left(-4,\text{\hspace{0.17em}}-31\right)$ .
If $x=-4$ and $y=-31$ , we get a true statement upon substitution and computataion.

$\begin{array}{lllll}\hfill y& =\hfill & 6x-7\hfill & \hfill & \hfill \\ -31\hfill & =\hfill & 6\left(-4\right)-7\hfill & \hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ -31\hfill & =\hfill & -24-7\hfill & \hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ -31\hfill & =\hfill & -31\hfill & \text{}\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

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
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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.