The system of equations shown is solved using the linear combination method
[tex]
\begin{array}{l}
6 x-5 y=-8 \quad \rightarrow \quad 6 x-5 y=-8 \quad \rightarrow \quad 6 x-5 y=-8 \
-24 x+20 y=32 \rightarrow \frac{1}{4}(-24 x+20 y=32) \rightarrow \frac{-6 x+5 y=8}{0=0}
\end{array}
[/tex]
What does [tex]$0=0$[/tex] mean regarding the solution to the system?
A. There are no solutions to the system because the equations represent parallel lines.
B. There are no solutions to the system because the equations represent the same line.
C. There are infinitely many solutions to the system because the equations represent parallel lines.
D. There are infinitely many solutions to the system because the equations represent the same line.
Answer (2)
The linear combination method results in the equation 0 = 0 .
The equation 0 = 0 indicates that the two original equations are dependent and represent the same line.
Since the equations represent the same line, there are infinitely many solutions to the system.
Therefore, the final answer is that there are infinitely many solutions to the system because the equations represent the same line. There are infinitely many solutions to the system because the equations represent the same line.
Explanation
Understanding the Result We are given a system of two equations that, after applying the linear combination method, results in the equation 0 = 0 . This outcome provides crucial information about the nature of the system's solutions.
Interpreting the Equation The equation 0 = 0 is always true, regardless of the values of x and y . This indicates that the two original equations are dependent, meaning one equation is a multiple of the other. In simpler terms, they represent the same line.
Determining the Solution Set When two equations represent the same line, every point on that line is a solution to both equations. Therefore, there are infinitely many solutions to the system.
Conclusion Thus, 0 = 0 means that there are infinitely many solutions to the system because the equations represent the same line.
Examples
Imagine you're trying to find the intersection of two roads on a map. If the equations representing the roads are dependent, it means the roads overlap completely. Every point on the overlapping section is a solution, representing infinitely many possible meeting points. This concept applies in various fields, such as economics, where dependent equations might indicate multiple equilibrium points, or in physics, where they could describe different states of a system that are fundamentally the same.
The equation 0 = 0 indicates that the equations represent the same line, leading to infinitely many solutions. Therefore, the answer is that there are infinitely many solutions to the system because the equations represent the same line. The correct choice is D.
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