Systems of Linear Equations and Inequalities

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What is a system of linear equations?

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A system of linear equations is a set of two or more linear equations with the same variables. The solution is the point(s) where the equations intersect.

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How do you determine if a point is a solution to a system of equations?

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To determine if a point is a solution, substitute the x and y values of the point into each equation. If the equations are satisfied, the point is a solution.

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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 equation?

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The graphical representation of a linear equation is a straight line on a coordinate plane, where each point on the line represents a solution to the equation.

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What is the difference between a linear equation and a linear inequality?

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A linear equation shows equality (e.g., y = mx + b), while a linear inequality shows a range of values (e.g., y < mx + b) and represents a region on a graph.

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What does the term 'feasible region' refer to in linear programming?

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The feasible region is the area on a graph that satisfies all constraints of a system of inequalities. It represents all possible solutions.

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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 if it were an equation, then shade the appropriate region that satisfies the inequality.

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

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The intersection point represents the solution to the system, where the values of the variables satisfy all equations simultaneously.

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What is the slope-intercept form of a linear equation?

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The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

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How can you tell if a system of equations has no solution?

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A system of equations has no solution if the lines are parallel, meaning they have the same slope but different y-intercepts.