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

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What does it mean to solve a system of equations using substitution?

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Solving a system of equations using substitution involves solving one of the equations for one variable and then substituting that expression into the other equation.

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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 graphs of the equations, which represents the solution to the system.

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What does it mean if a system of equations has 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 if a system of equations has 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 do you determine if a point is a solution to a system of equations?

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To determine if a point is a solution to a system of equations, substitute the coordinates of the point into each equation. If the point satisfies all equations, it is a solution.

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What is the substitution method in solving systems of equations?

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The substitution method involves isolating one variable in one equation and substituting that expression into the other equation to solve for the remaining variable.

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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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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.

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How can you identify a consistent and inconsistent system of equations?

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A consistent system has at least one solution, while an inconsistent system has no solutions. Consistent systems can be either independent (one solution) or dependent (infinitely many solutions).