Which of the following is a correct equation for the line passing through the point [tex]\((-2, 1)\)[/tex] and having slope [tex]\(m = \frac{1}{2}\)[/tex]?

Check all that apply.
A. [tex]\(y - 1 = \frac{1}{2}(x + 2)\)[/tex]
B. [tex]\(y = -2x + \frac{1}{2}\)[/tex]
C. [tex]\(y = \frac{1}{2}x + 2\)[/tex]
D. [tex]\(x - 2y = -4\)[/tex]



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].