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Match the number of solutions with its visual representation:

Equations:

1. [tex]\(y = -2x + 1\)[/tex]
2. [tex]\(2x + y = 1\)[/tex]

Choose:

A. No Solutions
B. One Solution
C. Infinitely Many Solutions

Sagot :

GinnyAnsweridn
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]
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