1.1 Introduction to Systems of Linear Equations Flashcards
Write a linear equation in two variables that would define a line in two dimensional space
a1x + a2y = b
a1, a2, b are constants
Write the equation of a plane in three dimensional space
a1x + a2y +a3z = b
coefficient
real numbers multiplied by variables next to them (denoted by a1, a2, … , an in linear equation)
constant term
real number denoted by b in a linear equation
leading coefficient
denoted as x1 in linear equation
leading variable
denotes as a1 in linear equation
Variables only appear to the ____ power in a linear equation.
first
(T/F) Linear equations can have products or roots of variables
False
(T/F) Linear equations do not have variables in trigonometric, exponential, or logarithmic functions
True
Write a linear equation in n variables
a1x1 + a2x2 +a3x3 + … + anxn = b
Define solution of linear equation
sequence of real numbers that satisfy the equation when you substitute the values in
Define solution set
set of all solutions of a linear equation
What does it mean to have solved a linear equation
Found its solution set
What are parametric representations of solution sets used for
Used to describe entire solution set of a linear equation with infinite solutions
Which is the free variable in the following equation:
x1 = 4 - 2x2
x2, it can take on any real value independently of any other variable
What is the parameter in the following equation:
x1 = 4 - 2t
t , a parameter replaces a free variable in a representation of a solution set with infinite solutions
Define system of m linear equations in n variables
set of m equations, each of which is linear in the same n variables
System of linear equations is also know as a _____
linear system
A system of linear equations is ______ when it has at least one solution
consistent
A linear system is ______ when it has no solutions. Give an example of such a system
inconsistent, for example two parallel lines
Describe row-echelon form. How do you solve a system in this form?
linear system arranged in a “stair-step” pattern with leading coefficients of 1. Solve using back-substitution.
Systems of linear equations are _____ when they have the same solution set
equivalent
List the operations that produce equivalent systems
- Interchange two equations.
- Multiply an equation by a nonzero constant.
- Add a multiple of an equation to another equation.
Creating a chain of equivalent systems using the 3 basic operations to rewrite the system of linear equations is called ________
Gaussian elimination
How do you recognize an inconsistent system during Gaussian elimination?
You obtain a false statement, such as 0 equaling a nonzero number
