Answer :
To determine the correct equations for the line that passes through the point (-2, 1) and has a slope of [tex]\( \frac{1}{2} \)[/tex], we will analyze each given option individually.
### Option A: [tex]\( y - 1 = \frac{1}{2}(x + 2) \)[/tex]
This equation is in point-slope form, which is given by [tex]\( y - y_1 = m (x - x_1) \)[/tex].
- Here, [tex]\( y_1 = 1 \)[/tex], [tex]\( m = \frac{1}{2} \)[/tex], and [tex]\( x_1 = -2 \)[/tex].
- Substituting these values, we get: [tex]\( y - 1 = \frac{1}{2}(x - (-2)) = \frac{1}{2}(x + 2) \)[/tex].
Therefore, option A is correct.
### Option B: [tex]\( v = -2x + \frac{1}{2} \)[/tex]
First, note that the equation is in the form [tex]\( v = \text{something} \)[/tex] instead of [tex]\( y = \text{something} \)[/tex]. Assuming "v" should be "y", let's analyze it:
- The given equation simplifies as [tex]\( y = -2x + \frac{1}{2} \)[/tex].
- The slope here is -2, which does not match the given slope of [tex]\( \frac{1}{2} \)[/tex].
Therefore, option B is incorrect.
### Option C: [tex]\( y = \frac{1}{2}x + 2 \)[/tex]
This equation is in slope-intercept form [tex]\( y = mx + b \)[/tex].
- Here, [tex]\( m = \frac{1}{2} \)[/tex], which matches the given slope.
- To check if this line passes through the point (-2, 1), substitute [tex]\( x = -2 \)[/tex] and see if [tex]\( y = 1 \)[/tex]:
[tex]\[ y = \frac{1}{2}(-2) + 2 = -1 + 2 = 1 \][/tex]
Therefore, it passes through the given point and option C is correct.
### Option D: [tex]\( x - 2y = -4 \)[/tex]
We will convert this equation to slope-intercept form [tex]\( y = mx + b \)[/tex].
- Start by isolating y:
[tex]\[ x - 2y = -4 \implies -2y = -x - 4 \implies y = \frac{1}{2}x + 2 \][/tex]
- Here, the slope is [tex]\( \frac{1}{2} \)[/tex], which matches the given slope, and the line passes through the point (-2, 1):
[tex]\[ y = \frac{1}{2}(-2) + 2 = -1 + 2 = 1 \][/tex]
Therefore, option D is correct.
### Summary
The correct options are:
- A. [tex]\( y - 1 = \frac{1}{2}(x + 2) \)[/tex]
- C. [tex]\( y = \frac{1}{2}x + 2 \)[/tex]
- D. [tex]\( x - 2y = -4 \)[/tex]
Thus, the correct options are [tex]\( \boxed{[1, 3, 4]} \)[/tex].
### Option A: [tex]\( y - 1 = \frac{1}{2}(x + 2) \)[/tex]
This equation is in point-slope form, which is given by [tex]\( y - y_1 = m (x - x_1) \)[/tex].
- Here, [tex]\( y_1 = 1 \)[/tex], [tex]\( m = \frac{1}{2} \)[/tex], and [tex]\( x_1 = -2 \)[/tex].
- Substituting these values, we get: [tex]\( y - 1 = \frac{1}{2}(x - (-2)) = \frac{1}{2}(x + 2) \)[/tex].
Therefore, option A is correct.
### Option B: [tex]\( v = -2x + \frac{1}{2} \)[/tex]
First, note that the equation is in the form [tex]\( v = \text{something} \)[/tex] instead of [tex]\( y = \text{something} \)[/tex]. Assuming "v" should be "y", let's analyze it:
- The given equation simplifies as [tex]\( y = -2x + \frac{1}{2} \)[/tex].
- The slope here is -2, which does not match the given slope of [tex]\( \frac{1}{2} \)[/tex].
Therefore, option B is incorrect.
### Option C: [tex]\( y = \frac{1}{2}x + 2 \)[/tex]
This equation is in slope-intercept form [tex]\( y = mx + b \)[/tex].
- Here, [tex]\( m = \frac{1}{2} \)[/tex], which matches the given slope.
- To check if this line passes through the point (-2, 1), substitute [tex]\( x = -2 \)[/tex] and see if [tex]\( y = 1 \)[/tex]:
[tex]\[ y = \frac{1}{2}(-2) + 2 = -1 + 2 = 1 \][/tex]
Therefore, it passes through the given point and option C is correct.
### Option D: [tex]\( x - 2y = -4 \)[/tex]
We will convert this equation to slope-intercept form [tex]\( y = mx + b \)[/tex].
- Start by isolating y:
[tex]\[ x - 2y = -4 \implies -2y = -x - 4 \implies y = \frac{1}{2}x + 2 \][/tex]
- Here, the slope is [tex]\( \frac{1}{2} \)[/tex], which matches the given slope, and the line passes through the point (-2, 1):
[tex]\[ y = \frac{1}{2}(-2) + 2 = -1 + 2 = 1 \][/tex]
Therefore, option D is correct.
### Summary
The correct options are:
- A. [tex]\( y - 1 = \frac{1}{2}(x + 2) \)[/tex]
- C. [tex]\( y = \frac{1}{2}x + 2 \)[/tex]
- D. [tex]\( x - 2y = -4 \)[/tex]
Thus, the correct options are [tex]\( \boxed{[1, 3, 4]} \)[/tex].
