Answer: y = 3x + 1 and y = -3x + 7
Step-by-step explanation: here's why: To determine which systems of equations have exactly one solution, we need to identify systems where the lines intersect at a single point. This point represents the unique solution to the system.
Let's analyze each system:
1. \(y = 3x + 1\) and \(y = -3x + 7\)
These two lines have different slopes. Therefore, they intersect at a single point, yielding one solution.
2. \(y = 3x + 1\) and \(y = x + 1\)
These lines have the same slope but different intercepts. Therefore, they are parallel and do not intersect. There is no solution.
3. \(y = 3x + 1\) and \(y = 3x + 1\)
These are the same equation. They represent the same line, so they have infinitely many solutions.
4. \(y = 3x + 1\) and \(y = 3x + 7\)
These lines have the same slope but different intercepts. They are parallel and do not intersect. There is no solution.
5. \(y = x + 1\) and \(y = 3x + 1\)
Same as case 2, they are parallel lines with different slopes and do not intersect. There is no solution.
6. \(y = 3x + 1\) and \(y = 3x + 7\)
Same as case 4, they are parallel lines with different intercepts and do not intersect. There is no solution.
7. \(x + y = 10\) and \(2x + 2y = 20\)
These two equations represent the same line. One equation is just a multiple of the other. They have infinitely many solutions.
8. \(x + y = 10\) and \(x + y = 12\)
These lines are parallel. They do not intersect, so there is no solution.
In summary, the systems of equations that have exactly one solution are:
- \(y = 3x + 1\) and \(y = -3x + 7\)
