Question

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.


WRITER

Answer 1

Part A:
System with no solution:
y = 2x + 3
y = 2x − 1

Explanation: Both equations have the same slope (2) but different y-intercepts (3 and −1), so the graphs are parallel lines. Parallel lines never intersect, so there is no point (x,y) that satisfies both equations — the system is inconsistent.

Part B:
Use the first equation from Part A, y = 2x + 3, and pair it with a line having a different slope, e.g. 3x − y = −4. The system is:
y = 2x + 3
3x − y = −4

Solve by substitution: substitute y = 2x + 3 into 3x − y = −4.
3x − (2x + 3) = −4
3x − 2x − 3 = −4
x − 3 = −4
x = −1

Now find y: y = 2(−1) + 3 = −2 + 3 = 1.

Solution: (x, y) = (−1, 1).