systems 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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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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What is the graphical representation of a system with an infinite number of solutions?

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The graphical representation of a system with an infinite number of solutions is when the equations represent the same line, meaning they overlap completely.

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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 other variable.

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

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You can determine this by analyzing the slopes and intercepts of the lines represented by the equations. If the slopes are different, there is one solution; if they are the same but different intercepts, there is no solution; if they are the same and the same intercept, there are infinitely many solutions.

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What is the solution to the system of equations 3x + 7y = 23 and -3x - 7y = -17?

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The solution is no solution, as the two equations represent parallel lines.

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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 an infinite number of solutions.

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What is the solution to the system of equations -4x - 2y = -12 and 4x + 8y = -24?

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The solution is (6, -6).

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What is the significance 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, affecting the number of solutions.

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How can you use substitution to solve a system of equations?

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You can solve a system of equations by solving one equation for one variable and substituting that expression into the other equation.