Questions and Answers
What are the 3 possible outcomes in a 2D linear system?
What is the form of a general solution?
(x, y)
What are the steps to solving a linear system in 2D?
- Multiply equations until same variable is each other's least common multiple. 2. Subtract equations to isolate a variable. 3. Plug isolated variable back into an original equation to find the other variable.
What do first degree equations refer to?
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How is a solution of an equation written?
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What is a consistent system?
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What is an inconsistent system?
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What does 'the system is m equations in n unknowns' mean?
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What is an equivalent system?
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What are the 3 ways of transforming systems?
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What does adding one equation to another look like?
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What are the 7 steps of the Echelon Method?
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X1, x2, x3 are another way to say what?
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What is the Echelon Method with no solutions?
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What is the Echelon Method with infinite solutions?
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What is the difference between general and particular solutions?
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0 = 0 is a true statement.
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0 = 20 is a true statement.
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Which variable is the parameter in the equation x = y + 3, with a general solution of (y + 3, y)?
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The original equations in a system can be solved by both particular and general solutions.
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How can you make the solution logical?
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Why does the system have infinite solutions for x + y + z = 40 and y + 3z = 49?
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Study Notes
Outcomes in a 2D Linear System
- Three possible outcomes: unique solution (lines intersect), no solution (parallel lines, inconsistent system), infinite solutions (lines are identical, equations are dependent).
General Solution
- The general solution of a linear equation, expressed as (x, y), indicates all possible solutions when there is more than one unique solution.
Steps to Solve a 2D Linear System
- Multiply equations to find the least common multiple for a variable.
- Subtract equations to isolate one variable.
- Substitute the isolated variable back to find the other variable.
Consistent and Inconsistent Systems
- A consistent system has at least one solution (could be unique or infinite).
- An inconsistent system has no solutions at all.
System of Equations
- The notation "m equations in n unknowns" indicates m = number of equations, n = number of variables.
Equivalent Systems
- Systems that share the exact same solution set; solutions must satisfy both systems to be considered equivalent.
Transforming Systems
- Three transforming operations:
- Exchange the order of equations.
- Multiply an equation by a non-zero real number.
- Add one equation to another.
Echelon Method Steps
- Confirm the presence of distinct variable terms in each equation.
- Eliminate terms systematically from the first equation onward.
- Formulate equations to simplify into the form Ay = K where A and K are constants.
- Ensure the leading coefficient of the first variable is 1 for reduced echelon form.
- Use substitution to solve each variable sequentially.
General vs. Particular Solutions
- General Solution replaces one variable with an expression in (x, y) form.
- Particular Solution arises from substituting a specific value into the general solution.
TRUE vs. FALSE Statements
- Examples: 0 = 0 is true, while 0 = 20 is false.
Parameters in Solutions
- In the equation x = y + 3, y acts as the parameter in its general solution (y + 3, y).
Existence of Solutions
- A system may show infinite solutions when there are more variables than equations, indicating degrees of freedom.
Logical Solutions
- When interpreting variables representing quantities (like workers), negative values are not applicable.
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Test your understanding of key concepts in Linear Algebra with these flashcards. This quiz covers outcomes of linear systems and the general solution form in a 2D context. Enhance your skills through engaging terminology and definitions.