Systems of Equations Substitution and Elimination Methods

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What is a system of equations?

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A set of two or more equations with the same variables.

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What does it mean to solve a system of equations?

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Finding the values of the variables that satisfy all equations in the system.

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What is the substitution method?

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A method for solving a system of equations by solving one equation for one variable and substituting that expression into the other equation.

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What is the elimination method?

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A method for solving a system of equations by adding or subtracting equations to eliminate one variable.

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What does it mean if a system of equations has no solution?

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The lines represented by the equations are parallel and never intersect.

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

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The equations represent the same line, meaning every point on the line is a solution.

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How do you determine if a system of equations has one solution, no solution, or infinitely many solutions?

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By analyzing the slopes and intercepts of the lines represented by the equations.

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Example of a system of equations: 2x + 3y = 6 and x - y = 1. Solve using substitution.

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(3, 0) is the solution.

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Example of a system of equations: 3x + 4y = 10 and 6x + 8y = 20. What can you conclude?

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Infinitely many solutions, as the second equation is a multiple of the first.

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What is the first step in the elimination method?

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Align the equations and manipulate them to have the same coefficient for one variable.