11.6: Summary of Key Concepts
Summary Of Key Concepts
System of Equations
A collection of two linear equations in two variables is called a system of equations.
Solution to a System
An ordered pair that is a solution to both equations in a system is called a solution to the system of equations. The values \(x=3,y=1\) are a solution to the system
\(\left\{\begin{array}{r}
x-y = 2\\
x + y = 4
\end{array}\right.\)
Independent Systems
Systems in which the lines intersect at precisely one point are independent systems. In applications, independent systems can arise when the collected data are accurate and complete.
Inconsistent Systems
Systems in which the lines are parallel are inconsistent systems. In applications, inconsistent systems can arise when the collected data are contradictory.
Dependent Systems
Systems in which the lines are coincident (one on the other) are dependent systems. In applications, dependent systems can arise when the collected data are incomplete.
Solving a System by Graphing
To solve a system by graphing:
- Graph each equation of the same set of axes.
- If the lines intersect, the solution is the point of intersection.
Solving a System by Substitution
To solve a system using substitution,
- Solve one of the equations for one of the variables.
- Substitute the expression for the variable chosen in step 1 into the other equation.
- Solve the resulting equation in one variable.
- Substitute the value obtained in step 3 into the equation obtained in step 1 and solve to obtain the value of the other variable.
- Check the solution in both equations.
- Write the solution as an ordered pair.
Solving a System by Addition
To solve a system using addition,
-
Write, if necessary, both equations in general form
\(ax+by=c\) - If necessary, multiply one or both equations by factors that will produce opposite coefficients for one of the variables.
- Add the equations to eliminate one equation and one variable.
- Solve the equation obtained in step 3.
- Substitute the value obtained in step 4 into either of the original equations and solve to obtain the value of the other variable.
- Check the solution in both equations.
- Write the solution as an ordered pair.
Substitution and Addition and Parallel Lines
If computations eliminate all variables and produce a contradiction, the two lines of the system are parallel and no solution exists. The system is inconsistent.
Substitution and Addition and Coincident Lines
If computations eliminate all variables and produce an identity, the two lines of the system are coincident and the system has infinitely many solutions. The system is dependent.
Applications
The five-step method can be used to solve applied problems that involve linear systems that consist of two equations in two variables. The solutions of number problems, mixture problems, and value and rate problems are examined in this section. The rate problems have particular use in chemistry.