Answer :
To solve the problem, we need to find the equation of the line that is parallel to the given line and passes through the point (4, 1).
1. Understand the definition of parallel lines:
- Parallel lines have the same slope but different y-intercepts.
2. Identify the slope of the given line:
- The given line is in point-slope form: [tex]\(y - 1 = -2(x - 4)\)[/tex].
- From this equation, we can see that the slope ([tex]\(m\)[/tex]) is [tex]\(-2\)[/tex].
3. Find the equation of a new line with the same slope passing through the given point (4, 1) using point-slope form:
- The point-slope form of a line equation is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
- Here, [tex]\( (x_1, y_1) = (4, 1) \)[/tex] and [tex]\( m = -2 \)[/tex].
4. Substitute the point and slope into the point-slope form equation:
- [tex]\(y - 1 = -2(x - 4)\)[/tex].
Thus, the equation of the line that is parallel to the given line and passes through the point (4, 1) is:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
Therefore, the correct equation is:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
1. Understand the definition of parallel lines:
- Parallel lines have the same slope but different y-intercepts.
2. Identify the slope of the given line:
- The given line is in point-slope form: [tex]\(y - 1 = -2(x - 4)\)[/tex].
- From this equation, we can see that the slope ([tex]\(m\)[/tex]) is [tex]\(-2\)[/tex].
3. Find the equation of a new line with the same slope passing through the given point (4, 1) using point-slope form:
- The point-slope form of a line equation is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
- Here, [tex]\( (x_1, y_1) = (4, 1) \)[/tex] and [tex]\( m = -2 \)[/tex].
4. Substitute the point and slope into the point-slope form equation:
- [tex]\(y - 1 = -2(x - 4)\)[/tex].
Thus, the equation of the line that is parallel to the given line and passes through the point (4, 1) is:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
Therefore, the correct equation is:
[tex]\[ y - 1 = -2(x - 4) \][/tex]
