What is a system of linear equations?

A collection of linear equations sharing variables; a solution satisfies them all at once.

What is a coefficient?

A number multiplying a variable in an equation, such as $-2$ in $5x-2y=11$.

What is a constant term?

A term without a variable, such as $11$ in $5x-2y=11$.

What does $A\mathbf{x}=\mathbf{b}$ represent?

The matrix form of a linear system: $A$ holds coefficients, $\mathbf{x}$ unknowns, and $\mathbf{b}$ constants.

How can you check a proposed solution?

Substitute its values into every original equation and verify that each equation is true.

Linear algebraLesson 1 of 2 in the course Systems of linear equations

Linear Systems: Solutions, Row Reduction, and Matrix Methods

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

A system of linear equations is a collection of equations that use the same variables. Solving a system means finding values that make every equation true at once.

A linear equation has variables only to the first power, with no products of variables. For example, $2x+3y=6$ is linear. An expression such as $xy=6$ is not linear because it multiplies variables.

All flashcards

What is a system of linear equations?: A collection of linear equations sharing variables; a solution satisfies them all at once.
What is a coefficient?: A number multiplying a variable in an equation, such as $-2$ in $5x-2y=11$ .
What is a constant term?: A term without a variable, such as $11$ in $5x-2y=11$ .
What does $A\mathbf{x}=\mathbf{b}$ represent?: The matrix form of a linear system: $A$ holds coefficients, $\mathbf{x}$ unknowns, and $\mathbf{b}$ constants.
How can you check a proposed solution?: Substitute its values into every original equation and verify that each equation is true.

Quiz questions

What must an assignment of values do to qualify as a solution to a system?Answer: It must satisfy every equation in the system simultaneously.
Which expression is not a linear equation?Answer: $xy=6$
Which operation is not an elementary row operation that preserves a system's solutions?Answer: Multiplying a row by zero
What form does Gaussian elimination produce by creating zeros below leading entries?Answer: Row echelon form
When is a system $A\mathbf{x}=\mathbf{b}$ consistent, according to the rank criterion?Answer: When $\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf{b}])$

Based on the Wikipedia article System of linear equations, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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