Question

Which of the following systems of linear equations has no solution?(1 point)
Responses

{4x+y=174x−y=3

{4x−y=174x−y=3

{4x−y=−174x+y=3

{4x−y=174x+y=3

Answer 1

To determine which of the given systems of linear equations has no solution, we can analyze each option for parallelism or inconsistency. A system of equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

Let's analyze each option:

1. **{4x + y = 17, 4x - y = 3}**
- Rewriting the equations in slope-intercept form (y = mx + b):
- Equation 1: y = -4x + 17
- Equation 2: y = 4x - 3
- The slopes are -4 and 4, which are different. **This system has a solution.**

2. **{4x - y = 17, 4x - y = 3}**
- Both equations are the same in terms of the left side:
- 4x - y = 17 (Equation 1)
- 4x - y = 3 (Equation 2)
- Since both cannot be true at the same time (one cannot equal two different constants), this system has no solution. **This system has no solution.**

3. **{4x - y = -17, 4x + y = 3}**
- Rewriting the equations:
- Equation 1: y = 4x + 17
- Equation 2: y = -4x + 3
- The slopes are 4 and -4, which are different. **This system has a solution.**

4. **{4x - y = 17, 4x + y = 3}**
- Rewriting the equations:
- Equation 1: y = 4x - 17
- Equation 2: y = -4x + 3
- The slopes are 4 and -4, which are different. **This system has a solution.**

In summary, the only option that has no solution is:
**{4x - y = 17, 4x - y = 3}**.