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 to the system is the set of values that satisfy all equations simultaneously.

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How can 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 point satisfies all equations, it is a solution.

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What does it mean for two equations to be consistent?

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Two equations are consistent if they have at least one solution. This can be one solution (intersecting lines) or infinitely many solutions (coincident lines).

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What does it mean for two equations to be inconsistent?

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Two equations are inconsistent if they have no solutions, meaning the lines represented by the equations are parallel and never intersect.

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What is the graphical representation of a system of equations?

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The graphical representation of a system of equations is the intersection of the lines represented by each equation. The intersection point(s) represent the solution(s) to the system.

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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 the significance of the coefficients in a system of equations?

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The coefficients in a system of equations determine the slope and position of the lines represented by the equations. They affect whether the lines intersect, are parallel, or are the same line.

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How do you find the y-coordinate of the solution in a system of equations?

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To find the y-coordinate of the solution, you can substitute the x-coordinate into one of the original equations and solve for y.

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What is an ordered pair in the context of systems of equations?

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An ordered pair is a pair of numbers (x, y) that represents a point on the Cartesian plane, which can be a solution to a system of equations.