systems practice w/applications

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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 do you represent a real-world problem using a system of equations?

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To represent a real-world problem using a system of equations, identify the variables that represent the unknowns, translate the relationships and conditions into equations, and ensure the equations reflect the problem's constraints.

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What does it mean to solve a system of equations graphically?

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Solving a system of equations graphically involves plotting the equations on a coordinate plane and finding the point(s) where the lines intersect, which represents the solution(s) to the system.

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What is the substitution method in 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 the variables.

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What is the elimination method in 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 dependent system of equations?

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A dependent system of equations has infinitely many solutions, as the equations represent the same line or plane in a geometric sense.

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What is an independent system of equations?

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An independent system of equations has exactly one solution, where the graphs of the equations intersect at a single point.

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What is an inconsistent system of equations?

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An inconsistent system of equations has no solution, as the equations represent parallel lines that never intersect.

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How can you check if your solution to a system of equations is correct?

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You can check if your solution is correct by substituting the values back into the original equations to see if they satisfy all equations.

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What role do coefficients play in a system of equations?

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Coefficients are the numerical factors in front of the variables in an equation. They determine the slope and position of the lines when graphed.