Certainly! Let's determine the number of solutions to the system of linear equations given.
We have two equations:
1. [tex]\( y = -2x + 1 \)[/tex]
2. [tex]\( 2x + y = 1 \)[/tex]
Step-by-Step Solution:
1. Write both equations in standard form:
- The first equation is already in slope-intercept form [tex]\( y = -2x + 1 \)[/tex].
- The second equation in standard form is [tex]\( 2x + y = 1 \)[/tex].
2. Compare the slopes and intercepts to determine the number of solutions:
- Let's rewrite the second equation in slope-intercept form to easily compare the slopes:
[tex]\( 2x + y = 1 \)[/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[ y = -2x + 1 \][/tex]
- Now the second equation [tex]\( y = -2x + 1 \)[/tex] is visually similar to the first equation [tex]\( y = -2x + 1 \)[/tex].
3. Visually represent the equations:
- If we plot these two lines on a graph, we notice that they are identical.
4. Determine the number of solutions:
- Since both equations represent the same line, every point on the line is a solution. Therefore, there are infinitely many solutions.
5. Conclusion:
Based on the comparison, we conclude:
- Infinitely many solutions.
So, the correct match for the number of solutions is:
[tex]\[ \boxed{Infinitely \; many \; solutions} \][/tex]
