1.1 Systems of Linear Equations

Question 1

In two dimensions a line in a rectangular xy-coordinate system can be represented by an equation of the form…

Answer 1

ax + by = c

a, b not both 0

Question 2

In three dimensions a plane in a rectangular xyz-coordinate system can be represented by an equation of the form…

Answer 2

ax + by + cz = d

a, b, c not all 0

Question 3

We say that ax + by = c is a _ equation in _.

Answer 3

linear equation in the variables x and y.

Question 4

We say that ax + by + cz = d is a _ equation in _.

Answer 4

linear equation in the variables x, y and z.

Question 5

Define a linear equation.

Answer 5

We define a linear equation in the n variables x1, x2, … , xn to be one that can be expressed in the form:
a1x1 + a2x2 + … + anxn = b
where a1, a2… an and b are constants and not all a’s are zero.

Question 6

What is a homogenous linear equation?

Answer 6

When its constant part is zero. i.e. where:

a1x1 + a2x2 + … + anxn = 0

Question 7

What do we call a linear equation where a1x1 + a2x2 + … + anxn = 0 ?

Answer 7

A homogeneous linear equation.

Question 8

What do we call a linear equation where its constant part is zero?

Answer 8

Homogeneous

Question 9

true / false

A linear equation can include products or roots of variables.

Answer 9

False.

A linear equation does not involve any products or roots of variables.

Question 10

true / false

A linear equation does not involve any products or roots of variables.

Answer 10

True

Question 11

true / false

In a linear equation all variables occur only to the first power.

Answer 11

True

Question 12

true / false

In a linear equation, variables may be raised to a power

Answer 12

False

In a linear equation all variables occur only to the first power.

Question 13

true / false

In a linear equation, variables may be arguments of trigonometric, logarithmic or exponential functions.

Answer 13

False

Question 14

true / false

In a linear equation, variables do NOT appear as arguments of trigonometric, logarithmic or exponential functions.

Answer 14

True

Question 15

A finite set of linear equations is called…

Answer 15

A system of linear equations, or more briefly,

a linear system

Question 16

What is a system of linear equations.

Answer 16

aka a ‘linear system’,

is a finite set of linear equations

Question 17

If a linear system has no solutions we say it is…

Answer 17

inconsistent

Question 18

If we say that a linear system is consistent, we mean…

Answer 18

that it has at least one solution

Question 19

Describe, in geometric terms, the possible solutions of a linear system of two unknowns.

Answer 19

The lines may be parallel and distinct (no intersection and no solution).
The lines may intersect at one point (exactly one solution).
The lines may coincide (infinitely many solutions).

Question 20

Every system of linear equations has … solutions.

Answer 20

either zero, one, or infinitely many solutions.

Question 21

A solution to a linear system in n unknowns is…

Answer 21

A sequence s1, s2 … sn where the substitution x1 = s1, x2 = s2 etc makes each and every equation in the system a true statement.
Solutions are usually written as: (s1, s2, … , sn) which is called an ordered n-tuple.

Question 22

We say that a linear system is consistent if…

Answer 22

…if it has at least one solution.

Question 23

We say that a linear system is inconsistent if…

Answer 23

…if it has no solutions.

Question 24

If we know a linear system to have at least one solution we say that the system is…

Answer 24

consistent

Question 25

If we know a linear system to have no solution we say that the system is…

Answer 25

inconsistent

Question 26

Every system of linear equations has how many solutions?

Answer 26

Zero, one, or infinitely many solutions. There are no other possibilities.

Question 27

A consistent linear system has how many solutions?

Answer 27

Either exactly one or infinitely many.

Question 28

What are the elementary row operations?

Answer 28

1. Multiply a row by a nonzero constant.
2. Interchange two rows.
3. Add a constant times one row to another.

Question 29

We say that two systems of equations are equivalent if…

Answer 29

…if they have the same solution set.

Question 30

If two systems of equations have the same solution set, we say that the systems are…

Answer 30

…are equivalent.