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<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

Solutions to linear equations in two variables

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 = 2 x + 5 .

  1. x = 4 , y = 13 ; since 13 = 2 ( 4 ) + 5 is true .
  2. x = 1 , y = 7 ; since 7 = 2 ( 1 ) + 5 is true .
  3. x = 0 , y = 5 ; since 5 = 2 ( 0 ) + 5 is true .
  4. x = 6 , y = 7 ; since -7 = 2 ( 6 ) + 5 is true .

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      ( a , b )

Ordered pair

In an ordered pair , ( a , b ) , the first component, a , gives the value of the independent variable, and the second component, b , gives the value of the dependent variable.

In an ordered pair (a,b), the first component a is the value of the independent variable, and the second component b is the value of the dependent variable.

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

( 0 , 7 ) .
If x = 0 and y = 7 , we get a true statement upon substitution and computataion.

y = 6 x 7 7 = 6 ( 0 ) 7 Is this correct? 7 = 7 Yes, this is correct .

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( 8 , 41 ) .
If x = 8 and y = 41 , we get a true statement upon substitution and computataion.

y = 6 x 7 41 = 6 ( 8 ) 7 Is this correct? 41 = 48 7 Is this correct? 41 = 41 Yes, this is correct .

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( 4 , 31 ) .
If x = 4 and y = 31 , we get a true statement upon substitution and computataion.

y = 6 x 7 31 = 6 ( 4 ) 7 Is this correct? 31 = 24 7 Is this correct? 31 = 31 Yes, this is correct .

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Questions & Answers

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RAW Reply
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Professor
I think
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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