18 Questions
What is the purpose of back-substitution in solving a system of linear equations?
To substitute values back into the original equations to solve for variables
When does a system of linear equations have no solution?
When it results in a false equation after elimination
What type of linear system is said to be dependent?
System with infinitely many solutions
In the context of linear systems, what does 'inconsistent' refer to?
Having no solutions
What is the purpose of Gaussian elimination in solving linear systems?
To eliminate variables through substitution
How does back-substitution help in finding solutions to a system of linear equations?
By substituting already solved variables back into the equations
What is the reason for eliminating variables in a system of linear equations?
To make the equations simpler
In Example 5, why did the process of adding the two equations lead to 0 = 29?
Inconsistent system
What is the equation of the line described in Example 6 in slope-intercept form?
$y = x - 2$
Why does an inconsistent system have no solution?
Due to conflicting equations
How are the lines in an inconsistent system, as mentioned in the text?
They are parallel and do not intersect
What does an infinitely many solutions scenario indicate for a system of linear equations?
The system is consistent
What is a system of linear equations?
A set of linear equations involving the same variables
How is a solution of a system of linear equations defined?
An assignment of values for the variables that make all equations true
In the Substitution Method for solving a system of linear equations with two variables, what is the first step?
Solve for one variable in terms of the other variable
What does it mean when a system of linear equations has no solution?
The system has no solutions that satisfy all the equations
When using the Elimination Method to solve a system of linear equations, what is the primary goal?
To eliminate one variable by adding or subtracting equations
What characterizes an inconsistent system of linear equations?
A system with no solutions that satisfy all equations
Study Notes
Solving Systems of Linear Equations
- A system of linear equations can be solved using Gaussian elimination to put it in triangular form, and then using back-substitution to find the solution.
- The solution of a system is an assignment of values for the variables that makes each equation in the system true.
Types of Solutions
- A system of linear equations in two variables has exactly one of the following:
- Exactly one solution
- No solution (inconsistent system)
- Infinitely many solutions (dependent system)
Example Systems
- Example 2: A system of linear equations with a unique solution (x = 3, y = 7, z = 4)
- Example 3: A system of linear equations with no solution (inconsistent system)
- Example 5: A system of linear equations with no solution (parallel lines)
- Example 6: A system of linear equations with infinitely many solutions (dependent system)
Methods for Solving Systems
- Substitution Method:
- Solve for one variable in terms of the other variable
- Substitute the expression into the other equation to obtain an equation in one variable
- Back-substitute to find the remaining variable
- Elimination Method:
- Multiply equations to eliminate a variable
- Add or subtract equations to eliminate the variable
- Solve for the remaining variables
Review of Systems of Linear Equations
- A system of linear equations is a set of linear equations that involve the same variables
- Objectives:
- Solve a system of linear equations using substitution, elimination, and graphical methods
- Determine when a system has one solution, no solution, or infinitely many solutions
Learn how to solve a system of equations using the elimination method. This quiz demonstrates the step-by-step process, from transforming the system to triangular form to using back-substitution to find the values of variables.
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