Solving Linear Systems by Various Methods

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What is a linear system?

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A linear system is a collection of two or more linear equations involving the same variables.

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What does it mean to solve a linear system?

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To solve a linear system means to find the values of the variables that satisfy all equations in the system simultaneously.

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What are the three main methods for solving linear systems?

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1. Graphing 2. Substitution 3. Elimination

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How do you solve a linear system by graphing?

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To solve by graphing, plot each equation on the same coordinate plane and identify the point(s) where the lines intersect.

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What is the substitution method?

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The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

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What is the elimination method?

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The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other.

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What does it mean if a linear system has no solution?

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A linear system has no solution if the equations represent parallel lines that never intersect.

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What does it mean if a linear system has infinitely many solutions?

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A linear system has infinitely many solutions if the equations represent the same line, meaning every point on the line is a solution.

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How can you identify a system with no solution?

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A system has no solution if, after manipulation, you end up with a false statement, such as 0 = 5.

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How can you identify a system with infinitely many solutions?

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A system has infinitely many solutions if, after manipulation, you end up with a true statement that involves a variable, such as 0 = 0.