7.5: Special Types of Linear Systems
Learning Objectives
At the end of this lesson, students will be able to:
- Identify and understand what is meant by an inconsistent linear system.
- Identify and understand what is meant by a consistent linear system.
- Identify and understand what is meant by a dependent linear system.
Vocabulary
Terms introduced in this lesson:
- consistent system
- inconsistent system
- infinite number of solutions
- dependent system
- determining the system type (graphically, algebraically)
Teaching Strategies and Tips
Use this lesson to classify systems of equations according to the number of solutions they have.
- Encourage students to use the graphical interpretation and rewrite equations in slope-intercept form first to compare slopes. See Examples 1-4.
Use Examples 5 and 6 to show how to classify a system using the substitution or elimination methods.
Solving inconsistent or dependent systems by substitution or elimination leads to variables dropping-out.
- A false statement, such as \begin{align*}3 = 4\end{align*}, indicates an inconsistent system (parallel lines).
- A true statement, such as \begin{align*}-12 = -12\end{align*}, indicates dependent system (coinciding lines).
When students arrive at an answer such as \begin{align*}8 = 8\end{align*}, they may assign \begin{align*}8\end{align*} to one of the variables and solve for the other variable.
Use Example 6 to point out that the equations in a dependent system are multiples of each other.
Additional Example:
Solve the system by multiplication.
\begin{align*}3x & = - y-5 \\ 2y + 10 & = -6x\end{align*}
Solution: Align the variables column-wise.
\begin{align*}3x+ y & = -5 \\ 6x +2y & = -10\end{align*}
Eliminate \begin{align*}y\end{align*} by multiplying the first equation by \begin{align*}-2\end{align*}.
\begin{align*}3x+ y = -5 && \xrightarrow{x(-2)} && -6x -2y = 10 \\ 6x +2y = -10 && \text{same} && 6x +2y = -10\end{align*}
Add.
\begin{align*}& -6x-2y = 10\\ & \quad 6x+2y \ = -10 \\ & \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ & \quad 0x + 0y \ = 0\end{align*}
Therefore, the system is dependent. Notice that the equations are multiples of one other: \begin{align*} 3x+ y = -5 \ \ \xrightarrow{x2} \ \ 6x +2y = -10.\end{align*}
In Examples 7-9, emphasize the geometric interpretation in context:
- In Example 7, the lines intersect because the rental fees are different for the two membership options.
- In Example 8, the lines are parallel because the memberships have different flat fees (\begin{align*}y-\end{align*}intercepts), but the same rental fee (slope).
- In Example 9, the lines coincide because the equations use the same information. It is not possible to determine the price of each fruit since the second equation does not give any new information.
Error Troubleshooting
General Tip: In dependent systems, students sometimes misinterpret “an infinite number of solutions” to mean that any ordered pair will satisfy the given system of equations. Of course, only the infinite number of points on the line that the equations represent are solutions.
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