Systems of Equations with Elimination and Substitution

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What is a system of equations?

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A system of equations is a set of two or more equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously.

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What are the two main methods for solving systems of equations?

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The two main methods for solving systems of equations are the elimination method and the substitution method.

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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 remaining variable.

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

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

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

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A system of equations 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 do you determine if a system of equations is consistent or inconsistent?

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A system is consistent if it has at least one solution; it is inconsistent if it has no solutions.

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What is the first step in the elimination method?

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The first step in the elimination method is to align the equations and manipulate them (by multiplying if necessary) so that the coefficients of one variable are opposites.

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How can you check your solution to a system of equations?

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You can check your solution by substituting the values back into the original equations to see if they hold true.

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What is a key advantage of the substitution method?

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A key advantage of the substitution method is that it can be easier to use when one equation is already solved for one variable.