\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Dependent Systems of Linear Equations - Infinitely Many Solutions

Algebra

Polynomials

Functions

Rational Functions

Graphing

Matrices

Systems

Trigonometry

Complex Numbers

Applications

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Articles

Algebra

Functions

Functions

Polynomials

Rational Functions

Graphing

Matrices & Systems

Matrices

Systems

Trigonometry & Complex Numbers

Trigonometry

Complex Numbers

Applications

SV Calculus

MV Calculus

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Articles

Types of Systems of Equations

Basics of linear systems with the same number of equations as unknowns - covered on the main linear systems page

Linear with fewer independent equations than unknowns - covered on this page

Non-Linear - covered on the non-linear system page

Consistent System With Infinite Solutions

When we have a (consistent) linear system with infinitely many solutions or we have a linear system with fewer equations than unknowns, we have what is known as a dependent system. These two cases are essential the same, since in a consistent system we have the same number of equations as unknowns but one of the equations does not give us any more information than one of the other equations. So we can eliminate the duplicate equation and then solve the system as if we had fewer equations than unknowns.

The key to this technique is to know how to describe the answer since we have infinitely many solutions. Many instructors just tell you to stop your work once you realize this but knowing how to describe the answer is a valuable technique. So we show you how to do this here.

Let's start with the case where you are solving a consistent linear system and you realize that you have infinitely many solutions. There are a couple of ways to know this based on which technique you are using. They are equivalent but they look a bit different.

1. If you are using substitution or basic elimination, you will add two equations together and get an equation with no variables which is true all the time, like \(3=3\) or \(0=0\).
2. If you are using gaussian elimination (also called row reduction), you will get a row with all zeros.
In both of these cases, you have eliminated the redundent information and you now have fewer equations than unknowns. Take a few seconds to write out your remaining equations and you will see that the next section can now be applied to your system.

Okay, time for some practice problems.

Practice

\(3x-2y+4z=1 \text{ and } x+y-2z=3 \text{ and } 2x-3y+6z=8\)

Problem Statement

\(3x-2y+4z=1 \text{ and } x+y-2z=3 \text{ and } 2x-3y+6z=8\)

Solution

1772 video

video by mattemath

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\(x+2y-7z=-4 \text{ and } 2x+3y+z=5 \text{ and } 3x+7y-36z=-25\)

Problem Statement

\(x+2y-7z=-4 \text{ and } 2x+3y+z=5 \text{ and } 3x+7y-36z=-25\)

Solution

1773 video

video by mattemath

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Sketch the graph of \(y=(1/3)^x \) and find the domain and range.

Problem Statement

Sketch the graph of \(y=(1/3)^x \) and find the domain and range.

Solution

2988 video

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Fewer Equations Than Unknowns - Non-Square Systems

At this point you have fewer equations than unknowns. That means there are infinitely many solutions to your system.

The first thing we do is to count the number of variables in the system and subtract the number of independent equations. This will give us the number of variables that we cannot account for. To learn how to handle this, let's watch this video. This guy explains the technique very well while he works a couple of examples.

Thinkwell - Nonsquare Systems [9min-14secs]

video by Thinkwell

So the idea is determine the number of variables that we cannot account for, usually just one, and solve the other variables in terms of the unknowns. It may seem strange to replace one variable with another one but this is the standard, so we will go with it.

Okay, time for some more practice problems.

Practice

\(x-y+z=-6 \text{ and } 2x+2y-6z=4\)

Problem Statement

\(x-y+z=-6 \text{ and } 2x+2y-6z=4\)

Solution

1768 video

video by Thinkwell

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\(4x-2y+6z=5 \text{ and } 2x-y+3z=2\)

Problem Statement

\(4x-2y+6z=5 \text{ and } 2x-y+3z=2\)

Solution

1769 video

video by Thinkwell

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\(x-y+4z=3 \text{ and } 4x-z=0\)

Problem Statement

\(x-y+4z=3 \text{ and } 4x-z=0\)

Solution

1771 video

video by mattemath

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\(12x+5y+z=0 \text{ and } 23x+2y-z=0\)

Problem Statement

\(12x+5y+z=0 \text{ and } 23x+2y-z=0\)

Final Answer

\((a/13, -5a/13,a)\)

Problem Statement

\(12x+5y+z=0 \text{ and } 23x+2y-z=0\)

Solution

2130 video

video by mattemath

Final Answer

\((a/13, -5a/13,a)\)

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\(9x+5y+9z=3 \text{ and } -45x+10y+27z=-6\)

Problem Statement

\(9x+5y+9z=3 \text{ and } -45x+10y+27z=-6\)

Solution

1770 video

video by PatrickJMT

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\(10x-6y+2z=0 \text{ and } 19x-5y-z=0\)

Problem Statement

\(10x-6y+2z=0 \text{ and } 19x-5y-z=0\)

Final Answer

\((a/4,3a/4,a)\)

Problem Statement

\(10x-6y+2z=0 \text{ and } 19x-5y-z=0\)

Solution

2131 video

video by mattemath

Final Answer

\((a/4,3a/4,a)\)

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Sketch the graph of \( y=2^x \) and find the domain and range.

Problem Statement

Sketch the graph of \( y=2^x \) and find the domain and range.

Solution

2987 video

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Sketch the graph of \( y = 3^{x-2} + 1 \) and find the domain and range.

Problem Statement

Sketch the graph of \( y = 3^{x-2} + 1 \) and find the domain and range.

Solution

2989 video

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Sketch the graph of \( y= 5 - 2^{3-x} \) and find the domain and range.

Problem Statement

Sketch the graph of \( y= 5 - 2^{3-x} \) and find the domain and range.

Solution

2990 video

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\(x+3y+2z=4 \text{ and } 2x+7y-z=5\)

Problem Statement

\(x+3y+2z=4 \text{ and } 2x+7y-z=5\)

Final Answer

\((-17z+13, 5z-3,z)\)

Problem Statement

\(x+3y+2z=4 \text{ and } 2x+7y-z=5\)

Solution

2132 video

Final Answer

\((-17z+13, 5z-3,z)\)

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Really UNDERSTAND Precalculus

Topics You Need To Understand For This Page

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

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