Final answer:
To create and solve a system of linear equations with one solution by elimination, we can set up two intersecting lines such as y = 2x + 3 and y = -x - 1, and eliminate variables to find the single point of intersection, which in this case is (x, y) = (-4/3, 1/3).
Step-by-step explanation:
To create a system of linear equations that has one solution and prove its uniqueness by solving with elimination, we can consider two equations that intersect at exactly one point. For instance
- Equation 1: y = 2x + 3
- Equation 2: y = -x - 1
To solve by elimination, we can multiply the second equation by 2 to match the coefficient of x in the first equation:
- Equation 2 modified: 2y = -2x - 2
Add Equation 1 and the modified Equation 2 to eliminate the y variable:
Now, solve for y:
Substituting y = 1/3 into Equation 1:
- 1/3 = 2x + 3
- 2x = -8/3
- x = -4/3
Thus, the solution to the system is (x, y) = (-4/3, 1/3), which proves that the system has only one solution.