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Systems of Linear Equations

Introduction

Consider the two equations ax+by=c and dx+ey=f. Since these equations represent two lines in the xy-plane, the simultaneous solution of these two equations (i.e. those points (x,y) that satisfy both equations) is merely the intersection of the two lines. The graphs below illustrate the three possible cases: non-parallel lines, parallel (but not identical) lines, and identical lines.

From left to right these cases yield one solution, no solutions, and infinite solutions. The same situation occurs in three dimensions; the solution of 3 equations with 3 unknowns is the intersection of the 3 planes.

There is a simple tool for determining the number of solutions of a square system of equations: the determinant.

It is often convenient to represent a system of equations as a matrix equation or even as a single matrix. Consider the system of equations 2x+3y=-8 and -x+5y=1. Since

we may write the entire system as a matrix equation:

or as AX=B where

The 2x2 matrix A is called the matrix of coefficients of the system of equations. Frequently this equation is written as a single augmented matrix:

We may now use Gaussian elimination to solve this matrix equation for x and y (as opposed to direct substitution of one equation into the other). This representation can also be done for any number of equations with any number of unknowns.

In general, the equation AX=B representing a system of equations is called homogeneous if B is the nx1 (column) vector of zeros. Otherwise, the equation is called nonhomogeneous.

Determining the Number of Solutions of a Nonhomogeneous System of Equations

For square systems of equations (i.e. those with an equal number of equations and unknowns), the most powerful tool for determining the number of solutions the system has is the determinant. Suppose we have two equations and two unknowns: ax+by=c and dx+ey=f with b and e non-zero (i.e. the system is nonhomogeneous). These are two lines with slope -a/b and -d/e, respectively. Let's define the determinant of a 2x2 system of linear equations to be the determinant of the matrix of coefficients A of the system. For this system

Suppose this determinant is zero. Then, this last equation implies a/b=d/e; in other words, the slopes are equal. From our discussion above, this means the lines are either identical (there is an infinite number of solutions) or parallel (there are no solutions). If the determinant is non-zero, then the slopes must be different and the lines must intersect in exactly one point. This leads us to the following result:

A nxn nonhomogeneous system of linear equations has a unique non-trivial solution if and only if its determinant is non-zero. If this determinant is zero, then the system has either no nontrivial solutions or an infinite number of solutions.

Example

The nonhomogeneous system of equations 2x+3y=-8 and -x+5y=1 has determinant

which implies the system has a unique solution (the point (-43/13,-28/75)). This corresponds to our intuition: the lines have different slopes and therefore must intersect in exactly one point.

Example

The nonhomogeneous system of equations x-4y+6z=3, -2x+8y-12z=-6, and 2x-y+3z=1 does not have a unique solution since its determinant is

In fact, this system has an infinite number of solutions. To see this, think geometrically. The three equations represent 3 planes. Two of the equations (the first and the second) represent the same plane ( why?); so, there are only two distinct non-parallel planes. Since two non-parallel planes intersect in a line, there are an infinite number of points which lie on all three of these planes (i.e. the system has an infinite number of solutions). To actually find the solutions, we must use a method such as Gaussian elimination to solve the system of equations.

Determining the Number of Solutions of a Homogeneous System of Equations

For a homogeneous system of equations ax+by=0 and cx+dy=0, the situation is slightly different. These lines pass through the origin. Thus, there is always at least one solution, the point (0,0). If the slopes -a/b and -c/d are equal then there are an infinite number of solutions since the lines are identical. But as we have seen, the slopes of these lines are equal when the determinant of the coefficient matrix is zero. Thus, for homogeneous systems we have the following result:

A nxn homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions.

Example

The system of equations 2x+y=0 and x-y=0 has exactly one solution since the slopes of the lines are different, i.e. the determinant

is non-zero. The solution is of course (0,0).

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Copyright © 1996 Department of Mathematics, Oregon State University

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Introduction Consider the two equations ax+by=c and dx+ey=f. Since these equations represent two lines in the xy-plane, the simultaneous solution of these two equations (i.e. those points (x,y) that satisfy both equations) is merely the intersection of the two lines. The graphs below illustrate the three possible cases: non-parallel lines, parallel (but not identical) lines, and identical lines.

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Scannable Document 10 on Jun 4, 2015, 9_46_32 AM

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The three types of solution sets:

A system of linear equations can have no solution, a unique solution or infinitely many solutions.

A system has no solution if the equations areinconsistent, they are contradictory. for example 2x+3y=10, 2x+3y=12 has no solution.

A system has aunique solution when it is consistent and the number of variables is equal to the number of nonzero rows.

A system hasinfinitely many solutions when it is consistent and the number of variables is more than the number of nonzero rows in the reference.

Zero: 2x+3y=10, 2x+3y=12

One solution: ( 2, 1, 3 ) or x=2, y=1, z=3

Infinite Solutions: has solution set (4-3z, 5+2z, z)

http://www.purplemath.com/modules/systlin2.htm

http://www.math.tamu.edu/~jlewis/Threetypesofsolsets.htm

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There are three types of solutions for linear equations.

One Solution:

(x=#)

In this type of solution which is the most common one, there are two lines which intersect only one time.

We identify this solutions by having the x equal to a number.

Zero:
(Ø)
This types of solutions are called inconsistent, because there is no intersection between lines.

we identify it when we have a 0 equal to a number

Infinit solutions:

(∞)

This type or solutions are infinite solutions, which are two lines that overlap because they have the same slope and are located in the same y intercept.

we identify this when both numbers are the same example 0=0

Stapel, E. (2015 April 1) System of linear equations systems: Graphing.

http://www.purplemath.com/modules/systlin2.htm

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'Type of Solution of Linear Equation'

-One Solution: this is the most common situarion and it involves lines that intersect exactly 1 time.

Example: x-2=4

6-2=4

x=6

-Zero: This only happens when the plines are parallel. Paralleel lines are not goong to ever meet.

Example:

6x+5-2x=4+4x+1

-Infinite Solution: this is the raresy case and only occurs when you have the same line.

Consider, the 2 lines below (y=2x+1 and 2y=4x+2). These twi equations are really the same line.

Example: line 1: y= 2x+1

Line 2 2y=4x+2

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There are 3 types of solutions:

one solution: or intersecting lines is when in a graph lines intersect once example: -7x+3=2x+2

answer:x=1/9

no solution: or inconsistent is when lines are parellel example: -7x+3=2x+2-9x

answer: 3=2

infinite solutions:or parallel lines is when linear are exactly the same example: -7x+2=2x+2-9x

answer: 0=0

https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/solutions-linear-equations/v/number-of-solutions-to-linear-equations

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One solution (=):Two distinct non-parallel lines that cross at exactly one point. This is called an "independent" system of equations, and the solution is always some x,y-point.

Zero solution (Ø): There is no solution because the original equation is a contradiction. The equations are called "inconsistent"

Infinite solution (∞): Appears to be only one line. But it's the same line drawn twice. These "two" lines, "intersect" at every point along their length. This is called a "dependent" system, and the "solution" is the whole line.

http://www.purplemath.com/modules/systlin2.htm

http://www.mathsisfun.com/algebra/images/system-linear-types.gif

https://www.khanacademy.org/math/algebra2/systems_eq_ineq/systems_tutorial_precalc/e/solutions_to_systems_of_equations

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1. One Solution: This is the most common situation and it involves lines that intersect exactly 1 time.

How to identify: If the lines only intercepts once in the graph.

Symbol: x={#}

Example:

2. Zero Solution: This only happens when the lines are parallel. They are never going to meet.

How to identify: The lines are parallel

Symbol: 0

Example:

3. Infinite Solution: This is the rarest case and only occurs when you have the same line.

How to identify: The lines are in the same place.

Symbol: x|

Example:

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Raul Perales, Federico de Zamacona Aguilar/Types of Solutions

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Raul Perales, Federico de Zamacona Aguilar/Types of Solutions

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Types of solutions

One solution: Two different non-parallel lines that cross at one point. They are called "INDEPENDENT"

Zero solution: Lines are parallel that meet. They are called "INCONSISTENT"

Infinite solution: Two lines that intercept at every coordinate. It's called "DEPENDENT"

http://www.purplemath.com/modules/systlin2.htm

http://www.mathsisfun.com/algebra/systems-linear-equations.html

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One or infinitely many solutions are called "consistent"

-One Solution:= The line only intercepts once.

EX.

3x+7=11x+15

-Zero SolutionØ:is called inconsistent if it has no solution. And the lines are parallel they never meet.

EX.

2x+3=2x-7

-Infinite Solutions ∞You actually have 2 lines but one on top of each other so they intercept in every coordinate.

EX.

2x+3=2x+3 http://www.charleston.k12.il.us/cms/Teachers/math/PreAlgebra/paunit5/L5-4.pdf http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/system/system.html

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Types of solutions of Linear Equations

One solution: it occurs when two different lines (no parallel) the solution always appears in a x,y point.

EX.: x=2 {2}, y=10{10}, z=-3 {-3}

Zero solution: when there are no variables (lines are parallel)

EX.: 4=6 Ø , 10=3 Ø, 0=7 Ø

Infinite solution: when the two lines intersect in all points.

Ex.: 5=5, 9=9, 3=3 xlx is all real numbers.

http://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/index.php

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One Solution: when the lines intercept just once. This lines are non parallel.

How to identify: the lines meet one time in the graph

Notation: { # }

Example: 2x+13=13

Infinite Solutions: when it looks like there is only one line, but there are 2 different lines, on top of each other, intercepting in every coordinate

How to identify: it appears to be only one line.

Notation: x|x is all real numbers

Example: 2x+3=2x+3

Zero Solutions: when the lines do not intercept, not even once

How to identify: parallel linesgoing to thebsame direction, without touching.

Notation: Ø

Example: 2x+3=2x+7

http://www.purplemath.com/modules/systlin2.htm

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A Linear Equation is an equation for a line.

One Solution: When the lines only intercept once. The lines are not parallel.

ex. 2x + 3 = 7

Zero Solution: No solution exists, often called inconsistent. The lines are parallel.

ex. x + 4 = x + 3

Infinite Solution: Two lines over lap, it is often called inconsistent. Opposite to one solution equations. The lines intercept in every coordinate.

Ex

2x = x + x

3 = 3

(x – 2)(x + 2) = x ² – 4

http://www.jamesbrennan.org/algebra/intro%20to%20algebra/solutions_of_algebraic_equations.htm

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Marian Huesca-Victoria Yesaki/Types of solution

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Marian Huesca-Victoria Yesaki/Types of solution

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One solution: when there is two lines of the graphic intercepts one time.

Symbol: x=#

Identify: When the lines intercepts just once.

Ex: 2(x-1)+8=4x-20

2x-2+8=4x-20

2x+6=4x-20

2x-4x=-20-6

-2x=-26

x=13

Zero Solution: When the lines do not intercept. There's no solution.

Symbol: 0=#

Identify:

The lines are parallel

Ex: 4x+y=8
8x+2y=−14

Infinite Solution:

First Move the slider around to try to find the value of k which makes the two lines parallel.

Symbol: 0=0

Ex:

{2x-y=2

5x−+k⋅y=5

Bibliography:

http://www.ms.uky.edu/~pkoester/teaching/Fall13/MA162/Geogebra/SystemsNoAndInf.html

https://www.khanacademy.org/math/algebra2/systems_eq_ineq/systems_tutorial_precalc/e/solutions_to_systems_of_equations

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Lino Corlay-Mario Orozco/ Types of Solution

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Lino Corlay-Mario Orozco/ Types of Solution

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One Solution: It is when lines only cross each other once, it is consistent. "Independent"

Zero: When there is no solution, equations are called "inconsistent"

Infinite Solution: Is when two lives overlap. "Dependent"

Bibliography:

http://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/index.php

http://www.mathsisfun.com/algebra/systems-linear-equations.html