Question
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. \[ \begin{array}{r}-3 x+4 y=-5 \\ -6 x+8 y=-10\end{array} \] Answer One Solution No Solutions
Ask by Boyd Alexander. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The system has infinitely many solutions.
Solution
The given system of equations is:
\[
\begin{cases}
-3x + 4y = -5 \\
-6x + 8y = -10
\end{cases}
\]
Let's analyze the system:
1. **Check for Proportionality:**
- Multiply the first equation by 2:
\[
2(-3x + 4y) = 2(-5) \implies -6x + 8y = -10
\]
- This result is exactly the same as the second equation.
2. **Interpretation:**
- Since the second equation is a multiple of the first, both equations represent the same line. This means that every solution to the first equation is also a solution to the second equation.
3. **Conclusion:**
- There are **infinitely many solutions** because the two equations describe the same line.
**Answer:** Infinitely many solutions.
Reviewed and approved by the UpStudy tutoring team
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Explain 
Simplify this solution Beyond the Answer
This system of equations actually has infinitely many solutions! If you look closely, the second equation is simply a multiple of the first one (it's obtained by multiplying the first equation by 2). So, they represent the same line in the coordinate plane, meaning any point along that line is a solution. When faced with similar situations, it's essential to check if one equation can be derived from the other by multiplying or adding/subtracting. If they’re dependent, as in this case, you can expect more than just one solution!
