Solving Systems

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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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What does it mean for a system of equations to have no solution?

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A system of equations has no solution when the equations represent parallel lines that never intersect.

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How can you determine if a system of equations has one solution, no solution, or infinitely many solutions?

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You can determine the number of solutions by analyzing the slopes and intercepts of the lines represented by the equations. If the lines intersect at one point, there is one solution; if they are parallel, there is no solution; if they are the same line, there are infinitely many solutions.

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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 or curves represented by the equations on a coordinate plane.

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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.

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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 does it mean for two equations to be dependent?

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Two equations are dependent if they represent the same line, resulting in infinitely many solutions.

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What does it mean for two equations to be independent?

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Two equations are independent if they represent different lines that intersect at exactly one point, resulting in a unique solution.

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How do you find the solution to a system of equations graphically?

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To find the solution graphically, plot both equations on the same coordinate plane and identify the point where the lines intersect.

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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, affecting the number of solutions.