To determine the number of solutions for the given system of linear equations, we need to analyze both equations and compare their slopes and y-intercepts.
The given system of linear equations is:
[tex]\[
\begin{array}{l}
y = -4x - 5 \\
y = -4x + 1
\end{array}
\][/tex]
Let's break this down step-by-step:
1. Identify the slopes of both equations:
- The slope of the first equation [tex]\( y = -4x - 5 \)[/tex] is [tex]\(-4\)[/tex].
- The slope of the second equation [tex]\( y = -4x + 1 \)[/tex] is also [tex]\(-4\)[/tex].
2. Compare the slopes:
- Both equations have the same slope of [tex]\(-4\)[/tex].
3. Identify the y-intercepts of both equations:
- The y-intercept of the first equation [tex]\( y = -4x - 5 \)[/tex] is [tex]\(-5\)[/tex].
- The y-intercept of the second equation [tex]\( y = -4x + 1 \)[/tex] is [tex]\(1\)[/tex].
4. Compare the y-intercepts:
- The y-intercepts are different ([tex]\(-5\)[/tex] and [tex]\(1\)[/tex]).
Since the slopes are the same but the y-intercepts are different, the lines represented by these equations are parallel. Parallel lines never intersect, so there is no point that satisfies both equations simultaneously.
As a result, the system of equations has:
No solution
So, you should match the given system of equations with "No solution".
