How many solutions exist for the system of equations below?
The Correct Answer and Explanation is:
We are given the system of equations:
- 3x+y=183x + y = 18
- 3x+y=163x + y = 16
To determine how many solutions exist, let’s analyze these two equations.
Both equations have the same left-hand side: 3x+y3x + y, but the right-hand sides are different (18 and 16). This suggests that the lines represented by these equations are parallel but have different y-intercepts.
Step-by-step Explanation:
Let’s subtract the second equation from the first: (3x+y)−(3x+y)=18−16(3x + y) – (3x + y) = 18 – 16 0=20 = 2
This is a contradiction. The result, 0=20 = 2, is false and shows that the system of equations has no solution.
Why?
Graphically, both equations represent straight lines. Since their left-hand sides (slopes) are the same, they are parallel lines. However, because their right-hand sides (y-intercepts) differ, they never intersect.
In a system of linear equations:
- If the lines intersect at exactly one point → One solution
- If the lines are the same (overlap completely) → Infinitely many solutions
- If the lines are parallel and distinct → No solution
This system falls into the third case.
Final Answer:
None
This means the system is inconsistent, and the two equations contradict each other. There is no pair of values for xx and yy that will satisfy both equations simultaneously.
Summary:
The equations 3x+y=183x + y = 18 and 3x+y=163x + y = 16 describe two parallel but different lines. Because they never cross, there is no point (x, y) that solves both equations at once.
✅ Correct answer: none
