Math 324: Linear Algebra1.1: Introduction to Systems of Linear Equations

Mckenzie West

Last Updated: September 4, 2019

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DefinitionA linear equation in n variables x1, x2, . . . , xn is anequation of the form

a1x1 + a2x2 + · · ·+ anxn = b

where a1, a2, . . . , an are real numbers called coefficients and bis a real number called the constant term.

We call a1 the leading coefficient and x1 the leading

variable.

Thought Question. What do you notice about these last twodefinitions?

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DefinitionA linear equation in n variables x1, x2, . . . , xn is anequation of the form

a1x1 + a2x2 + · · ·+ anxn = b

where a1, a2, . . . , an are real numbers called coefficients and bis a real number called the constant term.We call a1 the leading coefficient and x1 the leading

variable.

Thought Question. What do you notice about these last twodefinitions?

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DefinitionA linear equation in n variables x1, x2, . . . , xn is anequation of the form

a1x1 + a2x2 + · · ·+ anxn = b

where a1, a2, . . . , an are real numbers called coefficients and bis a real number called the constant term.We call a1 the leading coefficient and x1 the leading

variable.

Thought Question. What do you notice about these last twodefinitions?

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ExerciseWhich of the following equations are linear equations? For thelinear equations, identify the constant term, leading coefficient,and leading variable.

(a) x − 2y = 1

(b) sin x + cos y = 1

(c) x − xy + y = 3

(d) 5x1 = 6 + 3x2

(e) 3(x1+5) = 2(−6−x2)−2x3

(f)2z

z + 3=

3

z − 10+ 2

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DefinitionA solution to a linear equation in n variables is a sequence of nreal numbers s1, s2, . . . , sn arranged so that when you substitutethe values x1 = s1, x2 = s2, . . . , xn = sn the equation is satisfied.

ExerciseFind a solution to x − 2y = 1. (Be creative here.)How many solutions does this equation have? Can you describethem all?

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DefinitionA solution to a linear equation in n variables is a sequence of nreal numbers s1, s2, . . . , sn arranged so that when you substitutethe values x1 = s1, x2 = s2, . . . , xn = sn the equation is satisfied.

ExerciseFind a solution to x − 2y = 1. (Be creative here.)How many solutions does this equation have? Can you describethem all?

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DefinitionThe collection of all solutions of a linear equation is called asolution set. If you’re asked to solve a linear equation, youshould find all of the solutions and describe them.

Example

The solutions to the linear equation x + y = 4 are all of the formx = 4− y . We write these parametrically as y = s, x = 4− s.We call y a free variable because it is free to be whatever itwants to be.We call s a parameter and specific solutions can be found byassigning values to this parameter.

NoteTypically, leading variables are not used as free variables.

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DefinitionThe collection of all solutions of a linear equation is called asolution set. If you’re asked to solve a linear equation, youshould find all of the solutions and describe them.

Example

The solutions to the linear equation x + y = 4 are all of the formx = 4− y . We write these parametrically as y = s, x = 4− s.We call y a free variable because it is free to be whatever itwants to be.We call s a parameter and specific solutions can be found byassigning values to this parameter.

NoteTypically, leading variables are not used as free variables.

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DefinitionThe collection of all solutions of a linear equation is called asolution set. If you’re asked to solve a linear equation, youshould find all of the solutions and describe them.

Example

The solutions to the linear equation x + y = 4 are all of the formx = 4− y . We write these parametrically as y = s, x = 4− s.We call y a free variable because it is free to be whatever itwants to be.We call s a parameter and specific solutions can be found byassigning values to this parameter.

NoteTypically, leading variables are not used as free variables.

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ExerciseWrite all solutions to 5x = 6 + 3y in parametric form.

ExerciseWrite all solutions to 6x − 2y + 3z = 1 in parametric form. (Note:you may need more than one parameter.)

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ExerciseWrite all solutions to 5x = 6 + 3y in parametric form.

ExerciseWrite all solutions to 6x − 2y + 3z = 1 in parametric form. (Note:you may need more than one parameter.)

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DefinitionA system of m linear equations in n variables is a setof m equations each of which is linear in up to n variablesx1, . . . , xn.

Example

A system of 2 equations in 3 variables:

3x − 2y + 3z = 4x + z = 5

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DefinitionA system of m linear equations in n variables is a setof m equations each of which is linear in up to n variablesx1, . . . , xn.

Example

A system of 2 equations in 3 variables:

3x − 2y + 3z = 4x + z = 5

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DefinitionA solution to a system of m linear equations in n variables is anysequence of n real numbers that is a solution to every linearequation in the system. To solve a linear system is to find all ofthe solutions to the system.

A system is called consistent if it has at least one solution andinconsistent if it has none.

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DefinitionA solution to a system of m linear equations in n variables is anysequence of n real numbers that is a solution to every linearequation in the system. To solve a linear system is to find all ofthe solutions to the system.A system is called consistent if it has at least one solution andinconsistent if it has none.

Slide 9

ExerciseFind a solution to the system

3x − 2y + 3z = 4x + z = 5

ExerciseCome up with a linear system of 2 equations and 4 variables that isinconsistent.

Slide 9

ExerciseFind a solution to the system

3x − 2y + 3z = 4x + z = 5

ExerciseCome up with a linear system of 2 equations and 4 variables that isinconsistent.

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ExerciseFor each of the following systems, graph the two lines in thexy -plane. Where do they intersect? How many solutions does thissystem of linear equations have?

(a)3x − y = 12x − y = 0

(b)3x − y = 13x − y = 0

(c)3x − y = 16x − 2y = 2

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Thought Question. Is it possible for a system of linear equationsin two variables to have two solutions?

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ExerciseConsider the system of equations:

−2x − 2y + 2z − w = 22y − 2z + w = 0

2x = 2

Find all solutions to the system.

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ExerciseIn a chemical reaction, atoms reorganize in one or more substances.For instance, when methane gas (CH4) combines with oxygen (O2)and burns, carbon dioxide (CO2) and water (H2O) form. Chemistsrepresent this process by a chemical equation of the form

(x1)CH4 + (x2)O2 → (x3)CO2 + (x4)H2O.

Write a system of linear equations in the four variables x1, x2, x3, x4corresponding to the fact that the amount of each element–carbon,hydrogen, and oxygen–must be equal before and after the reaction.