Solving Systems of Equations

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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 is the set of values that satisfy all equations in the system 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 substitution and elimination.

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How does the substitution method work?

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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 remaining variable.

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What does it mean for a system of equations to be consistent?

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A consistent system of equations has at least one solution.

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What does it mean for a system of equations to be inconsistent?

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An inconsistent system of equations has no solutions, meaning the lines represented by the equations do not intersect.

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What is a dependent system of equations?

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A dependent system of equations has infinitely many solutions, meaning the equations represent the same line.

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

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You can determine this by analyzing the slopes and intercepts of the lines represented by the equations. If the slopes are different, there is one solution; if the slopes are the same but intercepts differ, there is no solution; if both slopes and intercepts are the same, there are infinitely many solutions.

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What is the first step in solving a system of equations by substitution?

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The first step is to solve one of the equations for one variable in terms of the other variable.

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When is it best to use the substitution method?

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The substitution method is best used when one equation is easily solvable for one variable.