Glossary
Algebraic Methods (Substitution, Elimination)
Techniques such as substitution or elimination used to find precise numerical solutions for systems of linear equations without graphing.
Example:
Using algebraic methods like substitution, you can solve and to find the exact solution .
Boundary Line
The line formed by replacing the inequality sign with an equals sign, which serves as the edge of the solution region for an inequality.
Example:
For the inequality , the line is the boundary line.
Bounded Regions
Solution regions for inequalities that have a finite area, typically forming a closed polygon on the coordinate plane.
Example:
The solution region for , , and is a bounded region forming a triangle.
Coincident (Overlapping) Lines
Lines that have the same slope and the same y-intercept, meaning they are the exact same line and represent a system with infinitely many solutions.
Example:
The equations and represent coincident lines.
Dashed Line
A type of boundary line used for strict inequalities (< or >), indicating that points on the line itself are not included in the solution.
Example:
When graphing , you would draw a dashed line to show the boundary is not part of the solution.
Graphical Method
A technique for solving systems of equations or inequalities by plotting their graphs and visually identifying intersections or shaded solution areas.
Example:
Using the graphical method, you can see that and intersect at (1,1).
Infinitely Many Solutions (for linear systems)
A scenario in a linear system where the lines are coincident, meaning every point on the line satisfies both equations.
Example:
The system and has infinitely many solutions as they are the same line.
Intersection Point (x, y)
A specific coordinate (x, y) where two or more lines cross on a graph, indicating the solution to a system of equations.
Example:
For the lines and , their intersection point is (2, 2).
Intersection of Shaded Regions
The overlapping area on a graph where the solution regions of multiple inequalities meet, representing the overall solution to the system of inequalities.
Example:
The area where the shading for and overlaps is the intersection of shaded regions.
No Solution (for linear systems)
A scenario in a linear system where the lines are parallel and never intersect, indicating that no point satisfies both equations.
Example:
The system and has no solution because the lines are parallel.
One Unique Solution
A type of solution for a linear system where the lines intersect at exactly one distinct point.
Example:
A system with lines and has one unique solution at (1, 2).
Parallel Lines
Lines that have the same slope but different y-intercepts, meaning they never intersect and thus represent a system with no solution.
Example:
The equations and represent parallel lines on a graph.
Shading (for inequalities)
The process of coloring the area on a graph that represents the solution set for a linear inequality or system of inequalities.
Example:
For , you would shade the region above the line to indicate all possible solutions.
Solid Line
A type of boundary line used for inclusive inequalities (≤ or ≥), indicating that points on the line itself are part of the solution.
Example:
When graphing , you would draw a solid line to include the boundary in the solution.
Solution (of a linear system)
The point(s) where the graphs of the equations in a system intersect, representing the values that make all equations true at the same time.
Example:
The point (1, 5) is a solution to the system and because it satisfies both equations.
Solution Region (for inequalities)
The area on a graph that satisfies all inequalities in a system simultaneously, often represented by a shaded area.
Example:
The shaded triangle formed by , , and is the solution region.
System of Linear Equations
Two or more linear equations that share the same variables, where the goal is to find values that satisfy all equations simultaneously.
Example:
The set of equations and forms a system of linear equations.
System of Linear Inequalities
Two or more linear inequalities sharing the same variables, defining an allowed region on a graph that satisfies all conditions.
Example:
The conditions and form a system of linear inequalities.
Test Points
Points chosen from different regions of a graph to check if they satisfy an inequality, used to confirm the correct shading.
Example:
To verify the shading for , you could use (0,0) as a test point to see if it satisfies the inequality (it shouldn't).
Unbounded Regions
Solution regions for inequalities that extend infinitely in one or more directions on the coordinate plane, having no finite area.
Example:
The solution for is an unbounded region extending infinitely to the right of the line .