System 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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What does it mean for a system of equations to be consistent?

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A consistent system of equations has at least one solution. It can be either a unique solution or infinitely many solutions.

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What does it mean for a system of equations to be inconsistent?

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An inconsistent system of equations has no solutions, meaning there is no set of values that satisfies all equations in the system.

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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. The point(s) of intersection represent the solution(s) to the system.

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How can you solve a system of equations using the substitution method?

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To solve a system using the substitution method, solve one equation for one variable and substitute that expression into the other equation, then solve for the remaining variable.

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How can you solve a system of equations using the elimination method?

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To solve a system using the elimination method, manipulate the equations to eliminate one variable by adding or subtracting the equations, then solve for the remaining variable.

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What is the solution to the system of equations y = 2x + 1 and y = 4x - 1?

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The solution is (1, 3), where both equations intersect.

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What is the solution to the system of equations y = 4x + 3 and 2x - 3y = 21?

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The solution is (-3, -9), found by substituting y from the first equation into the second.

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What is the importance of checking your solution in a system of equations?

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Checking your solution ensures that the values satisfy all equations in the system, confirming that the solution is correct.

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What is a unique solution in the context of a system of equations?

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A unique solution occurs when the system of equations has exactly one solution point where the lines intersect.