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How many solutions does this linear system have?

[tex]\[
\begin{aligned}
y & = -\frac{1}{2} x + 4 \\
x + 2y & = -8
\end{aligned}
\][/tex]

A. One solution: [tex]\((8, 0)\)[/tex]
B. One solution: [tex]\((0, 8)\)[/tex]
C. No solution
D. Infinite number of solutions


Sagot :

To determine the number of solutions for the given system of linear equations, we need to analyze their intersection, slope, and consistency.

Given equations:
[tex]\[ y = -\frac{1}{2}x + 4 \][/tex]
[tex]\[ x + 2y = -8 \][/tex]

### Step-by-Step Analysis:

1. Convert the equations to slope-intercept form (if necessary):

The first equation is already in slope-intercept form ([tex]\(y = mx + c\)[/tex]):
[tex]\[ y = -\frac{1}{2}x + 4 \][/tex]

The second equation needs to be converted. Start with:
[tex]\[ x + 2y = -8 \][/tex]

Isolate [tex]\(y\)[/tex]:
[tex]\[ 2y = -x - 8 \][/tex]
[tex]\[ y = -\frac{1}{2}x - 4 \][/tex]

2. Compare the slopes and y-intercepts:

- The first equation has a slope of [tex]\(-\frac{1}{2}\)[/tex] and a y-intercept of [tex]\(4\)[/tex].
- The second equation has the same slope of [tex]\(-\frac{1}{2}\)[/tex] but a different y-intercept of [tex]\(-4\)[/tex].

3. Interpret the results:

Since the slopes are identical ([tex]\(-\frac{1}{2}\)[/tex]) but the y-intercepts are different ([tex]\(4\)[/tex] and [tex]\(-4\)[/tex]), this indicates that the lines are parallel.

### Conclusion:
Parallel lines never intersect, meaning there are no points that satisfy both equations simultaneously.

Therefore, the system of equations has:

[tex]\[ \text{No solution} \][/tex]