1.1 Introduction to Systems of Linear Equations

Question 1

Write a linear equation in two variables that would define a line in two dimensional space

Answer 1

a1x + a2y = b

a1, a2, b are constants

Question 2

Write the equation of a plane in three dimensional space

Answer 2

a1x + a2y +a3z = b

Question 3

coefficient

Answer 3

real numbers multiplied by variables next to them (denoted by a1, a2, … , an in linear equation)

Question 4

constant term

Answer 4

real number denoted by b in a linear equation

Question 5

leading coefficient

Answer 5

denoted as x1 in linear equation

Question 6

leading variable

Answer 6

denotes as a1 in linear equation

Question 7

Variables only appear to the ____ power in a linear equation.

Answer 7

first

Question 8

(T/F) Linear equations can have products or roots of variables

Answer 8

False

Question 9

(T/F) Linear equations do not have variables in trigonometric, exponential, or logarithmic functions

Answer 9

True

Question 10

Write a linear equation in n variables

Answer 10

a1x1 + a2x2 +a3x3 + … + anxn = b

Question 11

Define solution of linear equation

Answer 11

sequence of real numbers that satisfy the equation when you substitute the values in

Question 12

Define solution set

Answer 12

set of all solutions of a linear equation

Question 13

What does it mean to have solved a linear equation

Answer 13

Found its solution set

Question 14

What are parametric representations of solution sets used for

Answer 14

Used to describe entire solution set of a linear equation with infinite solutions

Question 15

Which is the free variable in the following equation:

x1 = 4 - 2x2

Answer 15

x2, it can take on any real value independently of any other variable

Question 16

What is the parameter in the following equation:

x1 = 4 - 2t

Answer 16

t , a parameter replaces a free variable in a representation of a solution set with infinite solutions

Question 17

Define system of m linear equations in n variables

Answer 17

set of m equations, each of which is linear in the same n variables

Question 18

System of linear equations is also know as a _____

Answer 18

linear system

Question 19

A system of linear equations is ______ when it has at least one solution

Answer 19

consistent

Question 20

A linear system is ______ when it has no solutions. Give an example of such a system

Answer 20

inconsistent, for example two parallel lines

Question 21

Describe row-echelon form. How do you solve a system in this form?

Answer 21

linear system arranged in a “stair-step” pattern with leading coefficients of 1. Solve using back-substitution.

Question 22

Systems of linear equations are _____ when they have the same solution set

Answer 22

equivalent

Question 23

List the operations that produce equivalent systems

Answer 23

1. Interchange two equations.
2. Multiply an equation by a nonzero constant.
3. Add a multiple of an equation to another equation.

Question 24

Creating a chain of equivalent systems using the 3 basic operations to rewrite the system of linear equations is called ________

Answer 24

Gaussian elimination

Question 25

How do you recognize an inconsistent system during Gaussian elimination?

Answer 25

You obtain a false statement, such as 0 equaling a nonzero number