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

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What does it mean for a system of equations to have a unique solution?

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A system of equations has a unique solution if there is exactly one set of values for the variables that satisfies all equations in the system.

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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 if the equations represent parallel lines that never intersect.

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What does it mean for a system of equations to have infinitely many solutions?

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A system of equations has infinitely many solutions if the equations represent the same line, meaning every point on the line is a solution.

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

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You can determine this by analyzing the slopes and intercepts of the equations. If the slopes are different, there is a unique solution. If the slopes are the same but intercepts differ, there is no solution. If both slopes and intercepts are the same, there are infinitely many solutions.

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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 the 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 graphical method for solving systems of equations?

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The graphical method involves graphing each equation on the same coordinate plane and identifying the point(s) where the graphs intersect.

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What is the importance of the coefficients in a system of equations?

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The coefficients determine the slope and position of the lines represented by the equations, which affects the number of solutions.

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How do you interpret the solution of a system of equations in a real-world context?

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The solution represents the values of the variables that satisfy the conditions described by the equations, often corresponding to a specific scenario or problem.