Question
what is the best way to describe the solution of this system of equations?
2x - 5y = 8
5y = -2x + 8
2x - 5y = 8
5y = -2x + 8
Answers
GPT-4o mini
To describe the solution of the given system of equations:
1. **Identify the equations**:
- The first equation is \(2x - 5y = 8\).
- The second equation can be rearranged as \(5y = -2x + 8\) or \(y = -\frac{2}{5}x + \frac{8}{5}\).
2. **Graphical interpretation**:
- Each equation represents a line in the Cartesian plane. The solution to the system of equations is the point where the two lines intersect.
3. **Determine if the system is consistent and independent**:
- If the lines intersect at exactly one point, the system is consistent and independent, and has a unique solution. If the lines are parallel and never intersect, the system is inconsistent (no solution). If the lines are the same, there are infinitely many solutions.
4. **Solve the system**:
We can solve the first equation for \(y\):
\[
2x - 5y = 8 \implies 5y = 2x - 8 \implies y = \frac{2}{5}x - \frac{8}{5}
\]
Notice that this representation is identical to the second equation once rearranged, indicating they are indeed the same line.
Since both equations represent the same line, we conclude that:
- The system has infinitely many solutions, which can be expressed as any point \((x, y)\) that lies on the line defined by either equation.
**In conclusion**: The best way to describe the solution of this system of equations is that there are infinitely many solutions, represented by the line \(y = \frac{2}{5}x - \frac{8}{5}\).
1. **Identify the equations**:
- The first equation is \(2x - 5y = 8\).
- The second equation can be rearranged as \(5y = -2x + 8\) or \(y = -\frac{2}{5}x + \frac{8}{5}\).
2. **Graphical interpretation**:
- Each equation represents a line in the Cartesian plane. The solution to the system of equations is the point where the two lines intersect.
3. **Determine if the system is consistent and independent**:
- If the lines intersect at exactly one point, the system is consistent and independent, and has a unique solution. If the lines are parallel and never intersect, the system is inconsistent (no solution). If the lines are the same, there are infinitely many solutions.
4. **Solve the system**:
We can solve the first equation for \(y\):
\[
2x - 5y = 8 \implies 5y = 2x - 8 \implies y = \frac{2}{5}x - \frac{8}{5}
\]
Notice that this representation is identical to the second equation once rearranged, indicating they are indeed the same line.
Since both equations represent the same line, we conclude that:
- The system has infinitely many solutions, which can be expressed as any point \((x, y)\) that lies on the line defined by either equation.
**In conclusion**: The best way to describe the solution of this system of equations is that there are infinitely many solutions, represented by the line \(y = \frac{2}{5}x - \frac{8}{5}\).