Systems of Equations

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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 for a system of equations to have a unique solution?

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It means there is exactly one set of values for the variables that satisfies all equations in the system.

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What does it mean for a system of equations to have no solution?

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It means that the equations represent parallel lines that never intersect.

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What does it mean for a system of equations to have infinitely many solutions?

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It means that the equations represent the same line, resulting in an infinite number of intersection points.

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How can you determine if a system of equations has a unique 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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What is the graphical representation of a system of equations?

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The graphical representation is the intersection points of the lines corresponding to each equation.

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What is the substitution method in solving systems of equations?

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A method where one equation is solved for one variable, and that expression is substituted into the other equation.

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What is the elimination method in solving systems of equations?

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A method where equations are added or subtracted to eliminate one variable, making it easier to solve for the other.

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What is the role of the determinant in solving systems of equations?

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The determinant helps determine whether a unique solution exists for a system of linear equations.

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What is the solution to the system of equations 2x - y = 0 and 4x - y = 0?

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The solution is (0, 0), indicating that both equations represent the same line.