Part A: Create a system of linear equations with no solution. In two or more complete sentences, explain the specific characteristics that you included in each equation to ensure that the system would not have a solution.
Part B: Using one of the equations that you created in Part A, create a system of linear equations that has one solution (x,y). Use substitution to solve the system.
Part A: One possible set of linear equations with no solution is:
2x + 3y = 4
4x + 6y = 8
These equations are multiples of each other, so they describe the same line in the xy-plane. Therefore, there is no way to find a point that satisfies both of them simultaneously, and the system has no solution.
Part B: Using the first equation from Part A, we can create a system with one solution by adding a second equation that intersects the first line at a unique point. For example:
2x + 3y = 4
2x - y = 3
To solve this system by substitution, we can solve the second equation for y in terms of x:
2x - y = 3
y = 2x - 3
Then, we can substitute this expression for y into the first equation:
2x + 3(2x - 3) = 4
8x = 13
Solving for x, we get:
x = 13/8
Finally, we can substitute this value of x back into either equation to find y:
2(13/8) + 3y = 4
y = 11/24
Therefore, the system has a unique solution at (13/8, 11/24).