Graphing Systems of Inequalities and Equations

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What is a system of inequalities?

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A system of inequalities is a set of two or more inequalities that share the same variables. The solution is the set of all points that satisfy all inequalities in the system.

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How do you determine the number of solutions in a system of linear equations?

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To determine the number of solutions in a system of linear equations, analyze the lines represented by the equations: if they intersect at one point, there is one solution; if they are parallel, there are no solutions; if they are the same line, there are infinitely many solutions.

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What does it mean for a point to be a solution to a system of inequalities?

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A point is a solution to a system of inequalities if it satisfies all the inequalities in the system, meaning it lies in the region defined by the inequalities.

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What is the graphical representation of a linear inequality?

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The graphical representation of a linear inequality is a half-plane divided by the line that corresponds to the equation of the inequality. The line is dashed if the inequality is strict (>, <) and solid if it is inclusive (≥, ≤).

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How do you graph a system of inequalities?

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To graph a system of inequalities, first graph each inequality as a line, then shade the appropriate half-plane for each inequality. The solution region is where the shaded areas overlap.

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What is the significance of the intersection point of two lines in a system of equations?

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The intersection point of two lines in a system of equations represents the unique solution to the system, where both equations are satisfied.

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What is the difference between a solution to a system of equations and a solution to a system of inequalities?

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A solution to a system of equations is a specific point that satisfies all equations, while a solution to a system of inequalities is a region of points that satisfy all inequalities.

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What does the notation (x, y) represent in the context of graphing?

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The notation (x, y) represents a point in a two-dimensional Cartesian coordinate system, where 'x' is the horizontal coordinate and 'y' is the vertical coordinate.

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How can you verify if a point is a solution to a system of inequalities?

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To verify if a point is a solution to a system of inequalities, substitute the coordinates of the point into each inequality. If the point satisfies all inequalities, it is a solution.

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What is the role of the boundary line in graphing inequalities?

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The boundary line in graphing inequalities separates the two half-planes and indicates where the inequality holds true. It is dashed for strict inequalities and solid for inclusive inequalities.