Writing Systems of Equations and Inequalities

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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 simultaneously.

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How do you represent a system of equations graphically?

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A system of equations can be represented graphically by plotting each equation on the same coordinate plane. The point(s) where the lines intersect represent the solution(s) to the system.

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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. It can be either independent (one solution) or dependent (infinitely many solutions).

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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 solution. This occurs when the equations represent parallel lines that never intersect.

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What is an inequality?

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An inequality is a mathematical statement that compares two expressions, showing that one is less than, greater than, less than or equal to, or greater than or equal to the other.

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How do you represent a system of inequalities graphically?

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A system of inequalities is represented by shading the regions of the graph that satisfy each inequality. The solution set is where the shaded regions overlap.

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

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A linear equation represents a straight line and has an equal sign (=), while a linear inequality represents a region of the graph and uses inequality symbols (<, >, ≤, ≥).

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What is the substitution method for solving systems of equations?

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The substitution method involves solving one equation for one variable and then substituting that expression into the other equation to find the values of both variables.

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What is the elimination method for solving systems of equations?

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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 a feasible region in the context of inequalities?

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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.