7.2 Ordered Pair Solutions to Systems of Equations

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What is an ordered pair?

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An ordered pair is a pair of numbers used to represent a point in a two-dimensional space, written in the form (x, y), where 'x' is the horizontal coordinate and 'y' is the vertical coordinate.

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What does it mean for an ordered pair to be a solution to a system of equations?

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An ordered pair is a solution to a system of equations if it satisfies all equations in the system, meaning that when the values of 'x' and 'y' are substituted into the equations, both equations hold true.

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How do you determine if an ordered pair is a solution to a system of equations?

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To determine if an ordered pair is a solution, substitute the 'x' and 'y' values into each equation of the system. If both equations are true, then the ordered pair is a solution.

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What is the significance of the point of intersection in a system of equations?

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The point of intersection represents the solution to the system of equations, indicating the values of 'x' and 'y' that satisfy both equations simultaneously.

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What is the graphical representation of a system of equations?

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A system of equations can be represented graphically by plotting each equation on the same coordinate plane. The points where the graphs intersect represent the solutions to the system.

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What does it mean if two lines in a system of equations are parallel?

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If two lines in a system of equations are parallel, it means they have the same slope but different y-intercepts, indicating that there is no solution to the system as they never intersect.

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What does it mean if two lines in a system of equations coincide?

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If two lines in a system of equations coincide, it means they are the same line, indicating that there are infinitely many solutions to the system.

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What is the slope-intercept form of a linear equation?

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The slope-intercept form of a linear equation is given by y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

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How can you verify if a point is a solution to a linear equation?

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To verify if a point is a solution to a linear equation, substitute the x-value into the equation and check if the resulting y-value matches the y-value of the point.

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What is the role of substitution in solving systems of equations?

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Substitution involves solving one equation for one variable and then substituting that expression into the other equation to find the values of the variables.