Question
What does it mean for a system to have
one solution? Explain what it means graphically for a system to have one solution.
one solution? Explain what it means graphically for a system to have one solution.
Answers
GPT-4o mini
For a system of equations to have **one solution**, it means that there is exactly one set of values for the variables that satisfies all equations in the system simultaneously. In the context of linear equations, this typically indicates that the equations represent lines that intersect at a single point.
### Graphical Interpretation:
1. **Lines in a 2D System**: When dealing with two linear equations in two variables (e.g., \( y = mx + b \)), each equation represents a line on a Cartesian plane. If the two lines intersect at exactly one point, that point is the unique solution to the system.
2. **Unique Intersection**: This singular intersection point means that there exists one specific pair of \( (x, y) \) coordinates that satisfies both equations. Mathematically, if the lines representing the equations have different slopes, they will indeed cross at one point.
3. **Visual Representation**:
- If the lines are represented graphically and they intersect, the point of intersection can be determined by graphing the equations.
- For example, if the equations are \( y = 2x + 1 \) and \( y = -x + 4 \), the resultant graph will show two lines intercepting at one unique point—let's say \( (1, 3) \) in this case.
4. **Contrast with Other Situations**:
- **No Solution**: If the lines are parallel (same slope but different y-intercepts), they never meet and thus, there is no solution.
- **Infinitely Many Solutions**: If the two lines overlap completely (the same line), there are infinitely many solutions.
In conclusion, when we say a system has one solution, it means there is a single intersection point for the graphical representations of the equations, indicating that there is a unique combination of variable values that satisfies all equations in the system.
### Graphical Interpretation:
1. **Lines in a 2D System**: When dealing with two linear equations in two variables (e.g., \( y = mx + b \)), each equation represents a line on a Cartesian plane. If the two lines intersect at exactly one point, that point is the unique solution to the system.
2. **Unique Intersection**: This singular intersection point means that there exists one specific pair of \( (x, y) \) coordinates that satisfies both equations. Mathematically, if the lines representing the equations have different slopes, they will indeed cross at one point.
3. **Visual Representation**:
- If the lines are represented graphically and they intersect, the point of intersection can be determined by graphing the equations.
- For example, if the equations are \( y = 2x + 1 \) and \( y = -x + 4 \), the resultant graph will show two lines intercepting at one unique point—let's say \( (1, 3) \) in this case.
4. **Contrast with Other Situations**:
- **No Solution**: If the lines are parallel (same slope but different y-intercepts), they never meet and thus, there is no solution.
- **Infinitely Many Solutions**: If the two lines overlap completely (the same line), there are infinitely many solutions.
In conclusion, when we say a system has one solution, it means there is a single intersection point for the graphical representations of the equations, indicating that there is a unique combination of variable values that satisfies all equations in the system.