To solve the given system of equations:
[tex]\[
\begin{cases}
y = -3x - 5 \\
y = -3x + 6
\end{cases}
\][/tex]
we need to analyze each equation and determine their relationship.
1. Identify the slope and y-intercept of each line:
- The first equation, [tex]\( y = -3x - 5 \)[/tex]:
- Slope ([tex]\(m\)[/tex]) is [tex]\(-3\)[/tex]
- y-intercept ([tex]\(b\)[/tex]) is [tex]\(-5\)[/tex]
- The second equation, [tex]\( y = -3x + 6 \)[/tex]:
- Slope ([tex]\(m\)[/tex]) is [tex]\(-3\)[/tex]
- y-intercept ([tex]\(b\)[/tex]) is [tex]\(+6\)[/tex]
2. Compare the slopes and y-intercepts:
- Both equations have the same slope of [tex]\(-3\)[/tex]. This means the lines are parallel.
- However, their y-intercepts are different ([tex]\(-5\)[/tex] for the first line and [tex]\(+6\)[/tex] for the second line).
3. Interpret the result:
- Since the lines have the same slope but different y-intercepts, they are parallel lines.
- Parallel lines do not intersect. Therefore, there is no point [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.
4. Conclusion:
- There is no solution to this system because the lines are parallel and do not intersect.
Thus, the correct answer is:
[tex]\[
\text{no solution: } \{\}
\][/tex]
