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Solving Systems of Linear Equations
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Terms in this set (12)
Linear Equation
An equation of the form: ax+by+c=0
Graph of Linear Equation
A straight line in two-dimensional space
System of Linear Equations
Two or more linear equations
Solution of a Linear Equation
A set of all points that satisfy all of the linear equations in the system
Methods to Find the Solution of a System of Linear Equations
Substitution, Graphing, and Linear Combination
Substitution Method for Solving a Linear System of Equations (2 equations)
A method to find the solution of a system of linear equations that involves:
- Solving for one variable in the first equation
- Substitute the solution into the second equation
- Solve the second equation for the single remaining variable
- Plug the solution for that variable into the first equation and solve for the other variable
Graphing Method for Solving a Linear System of Equations (2 equations)
A method to find the solution of a system of linear equations that involves:
- Plot the first equation. (If necessary, convert the form to slope intercept or point slope form)
- Plot the second equation
- Identify the intersection. This is the solution
Linear Combination Method for Solving a Linear System of Equations (2 equations)
A method to find the solution of a system of linear equations that involves:
- Put both equations into general form (or preferred form)
- Choose one variable to eliminate
- Multiply one or both equations by appropriate factors so that both equations will have the same coefficient for that variable (though one may be negative)
- Add or subtract the equations, so that one variable drops out, leaving the other in the equation
-Solve for the variable in that equation
- Plug the solution into either equation to solve for the second variable
Three Possible Cases for the Solution Set
- Null set (no solutions)
- One point
-Infinite number of points
Graphical Meaning of a Null Solution Set to a Linear System
The lines are parallel (do not intersect)
Graphical Meaning of a One Point Solution to a Linear System
The lines intersect at one point on the coordinate plane
Graphical Meaning of an Infinite Point Solution to a Linear System
The two equations represent the same line
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