Answer :
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]
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]
