To solve this problem, we first need to understand the properties and equation format of parallel lines and how to use the given point. The given equation is:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
This is in point-slope form of the equation of a line, which is generally given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line.
### Step 1: Determine the Slope of the Given Line
The slope [tex]\( m \)[/tex] of the given line can be directly read from the given equation. Comparing with the point-slope form, we see that:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
By comparing, [tex]\( m = -2 \)[/tex].
### Step 2: Use the Point Through Which the New Line Should Pass
The new line that is parallel will have the same slope [tex]\( m \)[/tex]. The given point through which this new line must pass is [tex]\( (4, 1) \)[/tex].
### Step 3: Form the Point-Slope Equation for the New Line
Substitute the slope [tex]\( m = -2 \)[/tex] and the point [tex]\((4, 1)\)[/tex] into the point-slope form of the line equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( y_1 = 1 \)[/tex] and [tex]\( x_1 = 4 \)[/tex]. Substituting these values, we get:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
### Conclusion:
The equation of the line that is parallel to the given line [tex]\( y - 1 = -2(x - 4) \)[/tex] and passes through the point [tex]\((4, 1)\)[/tex] is:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
Looking at the choices provided:
1. [tex]\( y - 1 = -2(x - 4) \)[/tex]
2. [tex]\( y - 1 = -\frac{1}{2}(x - 4) \)[/tex]
3. [tex]\( y - 1 = \frac{1}{2}(x - 4) \)[/tex]
4. [tex]\( y - 1 = 2(x - 4) \)[/tex]
The correct equation is the first one:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
Hence, the correct choice is [tex]\( \boxed{1} \)[/tex].
