Here are the definitions and illustrations for the vocabulary terms you provided:
How many points (times) can two lines possibly intersect?
Two lines can intersect in three distinct ways:
- At one point: The lines are intersecting lines that cross each other at exactly one point. This represents a consistent system of linear equations.
- At no points: The lines are parallel and never meet. This represents an inconsistent system of linear equations.
- At infinitely many points: The lines coincide and lie on top of one another. This represents a dependent system of linear equations.
System of Linear Equations
A system of linear equations is a collection of two or more linear equations with the same variables. The solution to the system is the set of values for the variables that satisfy all equations in the system simultaneously.
Example:
- Equation 1: \( y = 2x + 3 \)
- Equation 2: \( y = -x + 1 \)
Graphically, this system would be represented by two lines on a coordinate plane.
Solution of a System as an Ordered Pair
The solution of a system of linear equations is an ordered pair (x, y) that satisfies all equations in the system. It represents the point of intersection of the lines represented by the equations.
Example: For the system:
- \( y = 2x + 3 \)
- \( y = -x + 1 \)
By solving these equations, you might find a solution such as \( (x, y) = (-2, -1) \). This means that the two lines intersect at the point (-2, -1) on the coordinate plane.
Illustration Summary
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Intersections:
- One intersection (crossing lines)
- No intersections (parallel lines)
- Infinite intersections (coinciding lines)
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System of Equations: Two equations represented graphically as lines on a graph.
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Solution as Ordered Pair: Point (x, y) where the lines intersect, providing values that satisfy both equations.