To determine which system of linear equations has no solution, we need to check for parallel lines, which occurs when the equations have the same slope but different y-intercepts.
Let's analyze each system:
-
System 1: \[ \begin{cases} 4x + y = 17 \ 4x - y = 3 \end{cases} \]
- Rearranging the equations:
- \(y = -4x + 17\) (first equation)
- \(y = 4x - 3\) (second equation)
- The slopes are \(-4\) and \(4\) (not parallel, so this system has a solution).
- Rearranging the equations:
-
System 2: \[ \begin{cases} 4x - y = 17 \ 4x - y = 3 \end{cases} \]
- Both equations are identical in form, the first can be represented as \(y = 4x - 17\) and the second as \(y = 4x - 3\).
- They have the same slope (\(4\)) but different y-intercepts (17 vs. 3), so these lines are parallel (no solution).
-
System 3: \[ \begin{cases} 4x - y = 17 \ 4x + y = 3 \end{cases} \]
- Rearranged forms:
- \(y = 4x - 17\)
- \(y = -4x + 3\)
- The slopes are \(4\) and \(-4\) (not parallel), so this system has a solution.
- Rearranged forms:
-
System 4: \[ \begin{cases} 4x - y = -17 \ 4x + y = 3 \end{cases} \]
- Rearranged forms:
- \(y = 4x + 17\)
- \(y = -4x + 3\)
- The slopes are \(4\) and \(-4\) (not parallel), so this system has a solution.
- Rearranged forms:
The only system with no solution is System 2:
\[ \begin{cases} 4x - y = 17 \ 4x - y = 3 \end{cases} \]