Answer :
To determine which equation represents a line that is parallel to [tex]\( y = \frac{1}{2} x - 2 \)[/tex] and passes through the point [tex]\((-8, 1)\)[/tex], we need to follow these steps:
### Step 1: Identify the Slope of the Given Line
The equation [tex]\( y = \frac{1}{2} x - 2 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. From this equation, we can see that the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
### Step 2: Understand the Condition for Parallel Lines
Lines that are parallel have the same slope. Therefore, the slope of our new line must also be [tex]\( \frac{1}{2} \)[/tex].
### Step 3: Apply the Point-Slope Form
The line passes through the point [tex]\((-8, 1)\)[/tex], so we can use the point-slope form of the equation of a line:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is [tex]\((-8, 1)\)[/tex] and [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Substitute the values into the point-slope form:
[tex]\[ y - 1 = \frac{1}{2}(x - (-8)) \][/tex]
[tex]\[ y - 1 = \frac{1}{2}(x + 8) \][/tex]
### Step 4: Simplify to Slope-Intercept Form
Solve for [tex]\( y \)[/tex] to convert this equation to slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[ y - 1 = \frac{1}{2}x + \frac{1}{2} \times 8 \][/tex]
[tex]\[ y - 1 = \frac{1}{2}x + 4 \][/tex]
[tex]\[ y = \frac{1}{2}x + 4 + 1 \][/tex]
[tex]\[ y = \frac{1}{2}x + 5 \][/tex]
### Step 5: Identify the Correct Option
The equation we derived is [tex]\( y = \frac{1}{2} x + 5 \)[/tex]. Now, we check the given options:
[tex]\[ \begin{array}{c} 1. \ y = \frac{1}{2} x + 5 \\ 2. \ y = \frac{1}{2} x - 9 \\ 3. \ y = -2 x - 7 \\ 4. \ y = -\frac{1}{2} x + 5 \\ 5. \ y = -2 x - 5 \\ \end{array} \][/tex]
The correct equation is:
[tex]\[ y = \frac{1}{2} x + 5 \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{1} \][/tex]
### Step 1: Identify the Slope of the Given Line
The equation [tex]\( y = \frac{1}{2} x - 2 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. From this equation, we can see that the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
### Step 2: Understand the Condition for Parallel Lines
Lines that are parallel have the same slope. Therefore, the slope of our new line must also be [tex]\( \frac{1}{2} \)[/tex].
### Step 3: Apply the Point-Slope Form
The line passes through the point [tex]\((-8, 1)\)[/tex], so we can use the point-slope form of the equation of a line:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is [tex]\((-8, 1)\)[/tex] and [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Substitute the values into the point-slope form:
[tex]\[ y - 1 = \frac{1}{2}(x - (-8)) \][/tex]
[tex]\[ y - 1 = \frac{1}{2}(x + 8) \][/tex]
### Step 4: Simplify to Slope-Intercept Form
Solve for [tex]\( y \)[/tex] to convert this equation to slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[ y - 1 = \frac{1}{2}x + \frac{1}{2} \times 8 \][/tex]
[tex]\[ y - 1 = \frac{1}{2}x + 4 \][/tex]
[tex]\[ y = \frac{1}{2}x + 4 + 1 \][/tex]
[tex]\[ y = \frac{1}{2}x + 5 \][/tex]
### Step 5: Identify the Correct Option
The equation we derived is [tex]\( y = \frac{1}{2} x + 5 \)[/tex]. Now, we check the given options:
[tex]\[ \begin{array}{c} 1. \ y = \frac{1}{2} x + 5 \\ 2. \ y = \frac{1}{2} x - 9 \\ 3. \ y = -2 x - 7 \\ 4. \ y = -\frac{1}{2} x + 5 \\ 5. \ y = -2 x - 5 \\ \end{array} \][/tex]
The correct equation is:
[tex]\[ y = \frac{1}{2} x + 5 \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{1} \][/tex]
